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QUANTUM COMPUTATION AND THE STABILIZER
FORMALISM FOR ERROR CORRECTION
DANIEL BULHOSA
Abstract. Computation carried out on machines that exploit the fundamen-
tal laws of quantum mechanics have the potential to expediting the solution
of certain classes of problems exponentially. Such a machine is known as a
quantum computer. Quantum error correction is a field of special interest as
it is a prerequisite for the physical implementation of a quantum computer.
In this paper we present the stabilizer formalism, which allows for the simple
description of error occurrence and correction. A reference for good examples
of applications is provided.
1. Introduction
Quantum computation promises to exponentially expediting the solution of many
difficult computational problems. Early conceptions of quantum information and
quantum computing include a seminal paper by Ingarden [3] written in 1976, and a
talk by Feynmann in 1981 at MIT describing a basic model for a quantum computer.
Deutsch introduced the first model for a universal quantum computer [4]. Shor’s
discovery of polynomial time algorithms for solving the discrete logarithm and prime
factorization algorithms gave the field of quantum computation its contemporary
impetus [5]. The latter is seen as very promising for practical applications since it
could be used to break widely-used public-key cryptosystems such as RSA. Grover
later introduced an algorithm capable of doing a search in an unsorted database in
square root time [6] (as opposed to linear time in a standard computer). Recent
research efforts have focused on quantum algorithms that would speed up matrix
inversion [7] and machine learning [8] amongst other applications.
One of the practical factors that limited the implementation of a quantum com-
puter was the lack of effective error correction schemes. An early concern was that
the nature of measurement in quantum theory would make it impossible to carry
out error correction [9]. In order to correct an error it must be detected first, and
researchers were concerned that the process of detecting an error through measure-
ment would destroy the information to be corrected. Later it was discovered that it
is possible to carry out certain classes of measurements that would allow for error
detection without destroying the quantum information [10] [11].
This paper presents a basic introduction to quantum computation. Then it in-
troduces the stabilizer sormalism, which utilizes concepts from algebra to create a
compact description of quantum error detection and correction. In Section 2 we
construct the general mathematical formalism for quantum computation, including
Date: May 14, 2015.
1
2 DANIEL BULHOSA
the tensor product, states and state spaces, gates, and measurement. In Section 3
we show a few examples of simple quantum information errors and how they are
corrected. In Section 4 we introduce the stabilizer formalism and how it can be used
to describe errors and their correction. Finally we make some conclusive remarks
in Section 5.
2. General Formalism for Quantum Computation
In this section we give a basic introduction to the mathematical foundations of
quantum computation theory. We first introduce tensor products, a mathematical
construct used use two vector spaces to create a new one. We then introduce the
basic formalism utilized in quantum computation theory.
2.1. Tensor Products. In order to formally develop the idea of a tensor product
we must use the idea of a free vector field. Let S be a set and let K be a field. We
define the free vector field FK(S) to be the set of maps f : S −→ K. We think of
each such function as describing the sum of symbols of S multiplied by elements of
K:
f ←→
s∈S
f(s)s
Note that we have not established any algebraic relations between the elements of s
so the summation map is not defined. This is why the formal definition in terms of
functions is necessary, but for the sake of intuition we use the summation notation.
For example, if S{a, b} and K = R then 1a + πb describes the formal sum f(a) = 1
and f(b) = π.
Now, let V and W be vector spaces both over the same field K. We define
the tensor product V ⊗ W to be the set FK(V × W) equipped with the following
equivalence relations:
• Addition in V : (v1, w)+(v2, w) ∼ (v1 +v2, w), where v1, v2 ∈ V and w ∈ W.
• Addition in W: (v, w1) + (v, w2) ∼ (v, w1 + w2), where v ∈ V and w1, w2 ∈
W.
• Scalar Multiplication: k(v, w) ∼ (kv, w) ∼ (v, kw), where k ∈ K.
The operations of addition and multiplication in V ⊗ W are those inherited from
FK(V ×W), which are a generalization of the summation and scalar multiplication
operations of standard vector fields. We use the common shorthand v ⊗ w for the
element (v, w) of V ⊗ W.
We can create operators on the tensor space V ⊗W from operators on V and W
separately. Let F be an operator on V and let G be an operator on W. We define
the tensor product of these two operators, denoted by F ⊗ G, to be the map:
QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION3
F ⊗ G : V ⊗ W −→ V ⊗ W
(F ⊗ G)(v, w) = (Fv, Gw)
We use the common shorthand Fv ⊗ Gw for the image of v ⊗ w under F ⊗ G.
Often we will deal with extremely large tensor products. For example, consider
a system constructed by taking the tensor product of ten identical vector spaces V .
Let F and G be operators on V and suppose that we wish to express an operator
that applies F to the first and third entries, and G to the tenth entry. In the
notation we have presently this operator would be expressed as:
F ⊗ I ⊗ F ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ G
Here I is the identity operator. This notation is incredibly cumbersome, so we
adopt a more convenient one. Let V be a vector space and consider taking its
tensor product with itself N times. For 1 ≤ j ≤ N and for any operator F on V
we define:
Fj = I ⊗ I ⊗ ... ⊗ F
at jth slot
⊗... ⊗ I
Using this notation we can write the operator previously described as,
F1F3G10
which is a significantly more compact description. This follows from the fact that
each operator acts on its corresponding slot independently.
2.2. States and State Spaces. The fundamental unit of quantum computation is
called the quantum bit, or ”qubit” for short, and it is analogous to the classical bit.
Mathematically however, the qubit is very different from the ordinary bit. Whereas
the ordinary bit takes on the values 0 and 1, the quantum bit takes on vector values
on the Bloch Sphere H:
H = {(x, y) ∈ C2
: |x|2
+ |y|2
= 1}
This definition is motivated by the physical properties of two-state quantum
systems1
[1] [13] , which are used to implement qubits physically. If some qubit has
value (x, y) we say this tuple is the state of the qubit. Suppose we denote the state
(x, y) by the label ψ, then we use the notations:
4 DANIEL BULHOSA
|ψ = (x, y) =
x
y
We use the following notation to denote the conjugate transpose of the state ψ:
ψ| = |ψ
†
= (x, y)†
= x∗
y∗
Here x∗
denotes the complex conjugate of x. We define the inner product of |ψ
and |φ by φ|ψ . This inner product induces the following norm:
ψ|ψ = x∗
y∗ x
y
1/2
= |x|2 + |y|2 = 1
Note that the set of conjugate transposes forms a vector space, which we denote by
H†
.
We use the tensor product to describe multiple qubit systems. The generaliza-
tion of H is a subset of (C2
)⊗N
, so we discuss that first.
Let (C2
)⊗N
denote the tensor product of C2
with itself N times. A generic ele-
ment of (C2
)⊗N
will be a linear combination of terms of the form |ψ1 ⊗ ... ⊗ |ψN
where |ψ1 , ..., |ψN ∈ C2
. Given this we define the set (C2
)⊗N†
as the set of linear
combinations of elements of the form ψ1| ⊗ ... ⊗ ψN |, where ψ1| , ..., ψN | ∈ (C2
)†
and (C2
)†
is the set transposes of elements of C2
.
The inner product on (C2
)⊗N
is induced by that on C2
through the following
rule:
( φ1| ⊗ ... ⊗ φN |)(|ψ1 ⊗ ... ⊗ |ψN ) := φ1|ψ1 · · · φN |ψN
Here the second line denotes scalar multiplication of the N resulting complex num-
bers. Along with the linearity of the inner product this equation entirely specifies
its action of any two elements of (C2
)⊗N
. Setting φi = ψi for each i yields the
formula for the norm. The conjugate transpose is linear and distributes over the
tensor product:
(|ψ1 ⊗ ... ⊗ |ψN )†
= |ψ1
†
⊗ ... ⊗ |ψN
†
= ψ1| ⊗ ... ⊗ ψN |
Having discussed the space C2
we can now define the set of states H: The states
of an N qubit system take values in the sets HN defined as:
1One common such system is a single electron. Electrons possess a property called spin, which
is can be interpreted as a magnetic dipole generated by the electron.
QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION5
HN = |ψ ∈ (C2
)⊗N
| ψ|ψ = 1
We denote the corresponding subset of (C2
)⊗N†
by H†
N .
Since the states under consideration are elements of (C2
)⊗N
for some N we can
treat them as vectors and add them together. However, we are only interested
in elements of HN so after summing in (C2
)⊗N
we must normalize the resulting
element. By normalizing, we mean that we must rescale the vector by a complex
number so that it will have unit norm. For example, suppose that we want to add
the orthonormal states |ψ and |φ together to create a new state. The new state
would take the form:
|ψ + |φ
√
2
∈ HN
Note the factor of
√
2 dividing the sum. It is easy to check that this normalized
state has norm equal to 1 by taking the inner product of it with itself and using
the linearity of the inner product and of the adjoint operator:
|ψ + |φ
√
2
†
|ψ + |φ
√
2
=
ψ| + φ|
√
2
|ψ + |φ
√
2
=
1
2
( ψ|ψ + 2Re( φ|ψ ) + φ|φ )
=
1
2
(1 + 0 + 1) = 1
We call a state that is the normalized sum of other states a superposition of the
latter states.
Finally, having defined the idea of addition of states we can exploit the fact that
HN is a subset of a vector space by utilizing the basis of the latter. Let:
|0 =
1
0
, and |1 =
0
1
Then any vector in C2
can be written as a linear combination of this, and thus so
can any element of H after appropriate normalization. It is a straightforward result
from algebra that if |0 and |1 is a basis for C2
, then the set of vectors,
B = {|ψ1 ⊗ ... ⊗ |ψN | |ψi ∈ {|0 , |1 } for each i}
6 DANIEL BULHOSA
forms a basis for (Q2
)⊗N
. Thus we can describe the elements of H⊗N
by taking
linear combinations of the elements of B and normalizing appropriately. As a con-
vention when using this basis we suppress the tensor notation and consolidate all
of the 0’s and 1’s. Thus, for example, we would denote |0 ⊗ |1 ⊗ |1 as |011 .
There are special multi-qubit states whose nature partly gives quantum compu-
tation its edge over classical computation. These states are called entangled states,
which we introduce now for reference. They are defined to be states in which the
state of each composite qubit cannot be described separately from the state of the
other qubits. Note that:
1
2
(|00 − |01 − |10 − |11 ) =
|0 + |1
√
2
⊗
|0 − |1
√
2
This is an example of an unentangled state, as we can describe the composite state
as the tensor product of states for the individual qubits. On the other hand the
state,
|01 + |10
√
2
does not admit such a factorization so it is entangled.
2.3. Gates. In classical information theory there exists the notion of a logic gate.
Mathematically, this is described by the action of some modular arithmetic opera-
tion on some set of bits. The notion of a gate is generalized in quantum information
theory to the notion of a quantum gate. We begin the discussion by considering H
and then generalize to HN .
Recall that an operator in a finite dimensional vector space is said to be Her-
mitian if F = F†
. Here F†
denotes the matrix we get by taking the transpose of
F and taking the complex conjugate of each entry. Now, it is a straightforward
consequence of the definition that the most general Hermitian operator F on C2
takes the form,
F =
a b∗
b c
where a, c are real and b is complex. Write b = x+iy for x, y real, and let w = (a+c)/
2 and z = (a − c)/2. Then we can re-express the general Hermitian matrix as:
QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION7
F =
w + z x − iy
x + iy w − z
= w
1 0
0 1
+ x
0 1
1 0
+ y
0 −i
i 0
+ z
1 0
0 −1
= wI + xX + yY + zZ
The matrices X, Y, and Z are known as the Pauli matrices, and any Hermitian
operator in C2
can be expressed as a linear combination of them with the identity.
It is easy to check that the square of any Pauli matrix is equal to the identity. Also,
note that ZX = iY so we can express Y in terms of the other Pauli matrices. It is
useful to note that the commutators and anti-commutators of these matrices obey
the following relations:
[X, Y ] = 2iZ, [Y, Z] = 2iX, [Z, X] = 2iY
{X, Y } = {Y, Z} = {Z, X} = 0
Thus each Pauli matrix either commutes or anti-commutes with the other Pauli
matrices. Finally, note that:
X |0 = |1 , Y |0 = i |1 , Z |0 = |0
X |1 = |0 , Y |1 = −i |0 , Z |1 = − |1
These relations will prove valuable as we will show any operation on a single quibit
can be described by the Pauli matrices.
In H we define a quantum operation to be any unitary operator U : H −→ H
that can be expressed in the form U = eitF
, where F is Hermitian and t ∈ R. This
condition is motivated by Schrodinger’s equation, which describes the dynamics of
quantum systems. The real variable t in the exponent describes the time the gate
has been applied to the qubit. We prove an incredibly useful result:
Theorem: Let F be a Hermitian matrix. Then we can express U = eitF
in terms
of the Pauli matrices and the identity.
Proof: Let F = wI + xX + yY + zZ, x = (x, y, z), and ˆx = x/||x||. Also let:
ˆx · ˆX = xX + yY + zZ
Note that for non-negative integer n:
8 DANIEL BULHOSA
(||x||ˆx · ˆX)2k
= (xX + yY + zZ)
2
k
= x2
X2
+ y2
Y 2
+ z2
Z2
+ xy{X, Y } + yz{Y, Z} + zx{Z, X}
k
= I
(ˆx · ˆX)2k+1
= (xX + yY + zZ)
2k
(xX + yY + zZ)
= xX + yY + zZ
= ˆx · ˆX
It is a well known identity that if A and B commute then eA+B
= eA
eB
. Thus we
have that:
U = eitwI
eit||x||ˆx· ˆX
= eitw
∞
n=0
(itF)n
n!
= eitw
∞
k=0
(itF)2k
(2k)!
+
∞
k=0
(itF)2k+1
(2k + 1)!
= eitw
∞
k=0
(it||x||ˆx · ˆX)2k
(2k)!
+
∞
k=0
(it||x||ˆx · ˆX)2k+1
(2k + 1)!
= eitw
I
∞
k=0
(−1)k
(t||x||)2k
(2k)!
+ i(ˆx · ˆX)
∞
k=0
(−1)k
(t||x||)2k+1
(2k + 1)!
= eitw
I cos(t||x||) + (ˆx · ˆX)i sin(t||x||)
We define any quantum operation applied to a qubit intentionally to be a quan-
tum gate. This stands in contrast to unintentional quantum operations, which we
introduce in the next section. Note that the Pauli matrices are themselves unitary
operations, so they describe gates. The following gates, called the Hadamard and
phase gates, are important in quantum computation. We state them for reference:
T =
1
√
2
1 1
1 −1
, S =
1
√
2
1 0
0 i
Any operator on a vector space created by taking tensor products of spaces can
be described as the tensor product of operators on the tensored spaces. This fol-
lows by fixing all but one of the vectors in a tensor product. It follows that any
Hermitian operator on HN can be described as the tensor product of the Pauli
matrices. In turn, this implies that any quantum operation can be described as the
exponential of a tensor product of Pauli matrices.
We introduce another important gate for reference. This gate acts on two qubits
and it is called the C-NOT gate. It is analogous to the controlled NOT gate in
QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION9
classical information theory. Denote the C-NOT gate by C, then its action is
entirely specified by the following relations:
C |0 ⊗ |0 = |0 ⊗ |0 ,
C |1 ⊗ |0 = |1 ⊗ |0 ,
C |0 ⊗ |1 = |1 ⊗ |1 ,
C |1 ⊗ |1 = |0 ⊗ |1 ,
2.4. Measurement and State Collapse. Measurement is another major concept
in the field of quantum computation that has no analog in classical information the-
ory. Classical bits are commonly represented as voltages, and voltages can be en-
gineered in such a way that measuring them does not affect their value significantly.
In general, measuring a qubit changes its state. It is a postulate quantum me-
chanics2
that each measurable physical has associated a Hermitian operator F.
When we measure the physical value we retrieve an eigenvalue of this Hermitian
operator. The process of measurement changes the state, and the resulting fi-
nal state is the eigenvector corresponding to the eigenvalue. Thus the process of
measuring can be defined as a probability distribution of eigenvalues fi of F with
corresponding eigenvectors |fi :
Mψ,F : Eigenvalues(F) −→ [0, 1]
Mψ,F (fi) = | fi|ψ |2
For our purposes we can take the set of eigenstates to be finite. It is not obvious
that this operation is well defined as it is unclear that the probabilities add up to
one. This is the case however when the eigenvalues are distinct. Corollary 8.6.7(a)
from [14] guarantees that in vector spaces with inner products like that of HN the
eigenvectors of any Hermitian operator form an orthonormal basis.3
This being the
case we must have that:
2N
i=1
|fi fi|ψ = |ψ
=⇒
2N
i=0
| fi|ψ |2
=
2N
i=1
ψ|fi fi|ψ = ψ|ψ = 1
The eigenvalue that we measure determines the outcome of the random variable.
Thus if we measure fi the new state must be |fi . However, it is apparent that if
some of the eigenvalues are equal to each other, or degenerate, then the probabil-
ity map is not well defined since the final state itself will not well defined. This
problem is solved by generalizing the map in the following way: For j ≤ 2N let
f1, ..., fk denote the distinct eigenvalues of F, let mi denote the number of times
10 DANIEL BULHOSA
the ith eigenvalue is repeated, and let |fJ
i denote the jth orthonormal vector in
the subspace of eigenvalue fi. The generalized map is defined as:
Mψ,F : Eigenvalues(F) −→ [0, 1]
Mψ,F (fi) =
mi
j=1
| fj
i |ψ |2
If the eigenvalue fi is measured the resulting state is the projection of |ψ onto
the corresponding eigenspace:
|ψ −−−−−−−−→
fj
i Measured


mi
j=1
|fi fj
i |ψ




mi
j=1
| fj
i |ψ |2


The operator,
mi
j=1
|fj
i fj
i |
is the appropriate projection operator and the quotient on the right side of the
arrow simply guarantees that the final state is normalized to unity. Note that if
|ψ is an element of the eigenspace of fi then with probability 1 the eigenvalue fi
will be measured and the final state will |ψ itself.
3. Basic Quantum Error Correction
Let HK be a system of K qubits. We can map the state of the K-qubit system
injectively to a larger system HN , where N ≥ K. We call such a mapping an [N, K]
quantum code. Analogously to the classical information theory case, encoding our
information allows us to introduce redundancies in it that permit error correction.
An example of a simple quantum code is the bit flip code. Let |0 and |1 be our
logical states. The bit flip code is the map H1
−→ H3
such that:
|0 → |000 , |1 → |111
The states |000 and |111 are our logical states in the code. We define an error to
be an unintended change of our information state. This may occur due to noise in
the computational hardware or the information channel in which the information
2This axiom is supported by experimental evidence.
3Furthermore the theorem states that the eigenvalues of such an operator must be real. This is
important as the values that we retrieve when physically measuring an operator are the eigenvalues
of the operator. Physical devices as of yet only measure real numbered quantities.
QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION11
resides. We can describe any error as the action of a unitary operator on our state,
so any error be described in terms of the Pauli matrices and the identity matrix.
For example, a bit flip error can be described by the operation of an X operator
on one of the qubits:
|000 −−−−−→
X2 error
|010
Analogously to the classical case however, we can see that we can correct any such
single flip error by comparing all three qubit values and flipping the one in dis-
agreement.
One question naturally arises. Suppose that our erroneous information state is
a superposition of states:
|010 + |101
√
2
(1)
Intuition would suggest that in order to detect the error we need to measure the
states of the individual qubits. However measuring the individual qubits destroys
the superposition of states. Suppose for example that we measured the value of the
first qubit to be zero. This corresponds to measuring the Z1 operator, which leads
to the following transition:
|010 + |101
√
2
−−−−−−−−→
Measurement
|010
Superposition is a desirable feature of qubits as they allow for sophisticated
computations to be carried out in parallel. Thus is is necessary to come up with
measurements that do not destroy the quantum state. For this to be the case these
measurements must correspond to operators whose eigenstates are the erroneous
states we want to detect. Fortunately in most common cases such measurements
exist.
In this example measuring the operators Z1Z2, Z2Z3, and Z1Z3 determines ex-
actly which bit flip (if any) has occurred. If ZiZj is measured to be +1 qubits i
and j have the same value, whereas if −1 is measured they differ. In this example
we would measure −1, −1, and +1. This would imply that qubits 1 and 3 agree
with each other but qubit 2 disagrees with both of them. Thus qubit 2 was flipped.
Note that all of the possible erroneous single qubit flip states, including (1), are
eigenstates of these operators.
The bit flip error is familiar from classical error correction. There exists an error
in quantum computation that does not have a classical counterpart. This is the
phase flip error, which is described by the action of the operator Z. For example,
consider a phase flip on the second qubit:
12 DANIEL BULHOSA
|000 + |111
√
2
−−−−−→
Z2 error
|000 − |111
√
2
These two vectors are not the same and thus do not represent the same quantum
information state. However assuming only a phase flip error has occurred and that
no bit flip errors could have occurred we can correct this kind of error as we did in
the last example. Measurement of the operator X1X2X3 is sufficient.
The central insight in the study of quantum error correction is that any error
can be described in terms of bit flip and phase flip errors. This is a consequence of
the fact that any generic operation on a set of qubits can be described in terms of
the matrices X and Z. Since errors are simply unintended operations on quantum
informational states they admit this description.
4. Stabilizer Formalism
The stabilizer formalism provides a compact way of describing a large class of
error correction codes in quantum information theory. Its key realization realiza-
tion in is that we can use operators to uniquely identify certain quantum states.
Knowing this, we can describe quantum states in terms of a discrete set of operators
rather than as vectors on the Bloch sphere.
4.1. States. To illustrate how a state can be specified using operators, consider
the states |000 and |111 . Note that application of the operators Z1Z2 and Z2Z3
returns the same states. By using the matrix and vector representations of these
operators and vector respectively simple arithmetic can be used to show that up to a
global phase and normalization this is the unique vector satisfying these conditions.
Now we introduce the general group whose subsets we will utilize to describe
states. Let X, Y , and Z be the Pauli matrices, then we define the group GN to
be the set of all N-fold products of the Pauli matrices, allowing for multiplication
by ±1 and ±i. This group acts on systems of N qubits. For example, the tensor
product −iX1X2Y3Y4Z5 is an element of G5.
The subsets that we will utilize are called stabilizers. They are defined as follows:
Let S be a subset of Gn, and define VS to be the set of vectors mapped by every
element of S to themselves, or fixed by S. Then we say that S is the stabilizer of
VS, or that S stabilizes VS. Since the Pauli matrices are linear operators it is easy
to show that the set VS is a vector space. Returning to our original example the
subgroup,
{I1I2I3, Z1Z2, Z2Z3, Z1Z3}(2)
stabilizes the two-dimensional vector space spanned by |000 and |111 . Note that
X1Z1X2Z2 = −Y1Y2 so since X1X2 and Z1Z2 fix this state so does Y1Y2.
QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION13
It is often the case that a stabilizer will have many elements, so that specifying
it by specifying each element is not practical. However in algebra it is common
practice to specify a group in terms of its generators. Let G be some group and
let g1, ..., gn be elements of G. We say that the set {g1, ..., gn} generates G if every
element of G can be written as a product of elements of this set. We use the nota-
tion g1, ..., gn to denote the group generated by the set {g1, ..., gn}.
A given group does not necessarily have a unique set of generators. For example,
the group (2) is generated by both {Z1Z2, Z2Z3} and {Z1Z2, Z2Z3, Z1Z3}. However
the former set is smaller and thus more convenient. In general, for a given group we
always find at least one set of generators with minimal size. Such a set is called an
independent set of generators, and it is characterized by the condition that removing
any generator yields a different group:
g1, ..., gl−1, gl+1, ..., gn = g1, ..., gn for each 1 ≤ l ≤ n
Independent sets of generators provide the most compact description for a given
group, and thus we will utilize them when working with stabilizers.
4.2. Gates. We have devised a way of describing states in terms of sets of stabilizers
rather than vectors. Since we use the action of operators to describe changes in
states the question naturally arises: How do we describe the effect of an operator in
terms of stabilizers? The answer is in fact quite intuitive. Note that U is a unitary
operator and some operator W fixes |ψ then the operator UWU†
fixes U |ψ for:
(UWU†
)U |ψ = (UW)(U†
U) |ψ = UW |ψ = U |ψ
This makes it clear that if S is the stabilizer for the state |ψ then,
USU†
:= {UWU†
| W ∈ S}
is the stabilizer for W. Thus the action of a unitary operation U on a state repre-
sented by S is conjugation by U.4
In the stabilizer formalism the action of the basic quantum gates introduced in
Section 2.3 of the paper is simple. For example, the action of the Hadamard gate
on the Pauli matrices is:
HXH†
= Z, HY H†
= −Y, HZH†
= X
Though not all possible gates can be described as combinations of these basic gates,
most conventional quantum error corrections schemes can be described in terms of
4If W and U are operators we say that UWU† is the conjugate of W with respect to U. If S
is a set of operators, we call USU† the conjugate set to S with respect to U.
14 DANIEL BULHOSA
them. Furthermore it can be shown that any unitary operation mapping elements
of Gn to Gn can be described in terms of the Hadamard, phase, and C-NOT gates
[1]. Thus within the framework of the stabilizer formalism these basic gates allow
for a simple but complete description of the error correction process.
4.3. Measurement. Just as the process of measurement changes the state vector
of a system it also changes its stabilizer. The change that the stabilizer undergoes
depends on the relation of the observable g being measured to the generators of the
stabilizer of the state under consideration.
Suppose that the state under consideration |ψ has a stabilizer S = g1, ..., gn ,
where this set of generators is independent. Without loss of generality we can
assume that g is an element of GN with no multiplicative factor of ±i or −1 in
front of it. This assumption can be made as multiplication by an overall factor
does not change the eigenvalues and eigenvectors of g. This being the set-up of our
measurement problem, we have two possibilities:
• The observable g commutes with every element of S. In this case we have
for each gi ∈ S that gi(g |ψ ) = ggi |ψ = g |ψ . Thus g |ψ is stabilized
by S and thus it must be parallel to |ψ . Furthermore, since g is a tensor
product of Pauli matrices we must have that g2
= I. These two facts com-
bined imply that g |ψ = ± |ψ .
Thus we have shown that |ψ is an eigenvector of g in this case. Recall
that measurement of an operator for a state that is in an eigenspace of the
former leaves the latter unchanged. Thus the final state has an identical
stabilizer.
• The observable g anti-commutes with a single element of S. Note that this
is entirely general: For Example, g does not commute with two elements
g1 and g2 of S, then g does commute with g1g2:
[g1g2, g] = g1g2g − gg1g2
= −g1gg2 − gg1g2
= g(g1g2 − g1g2) = 0
This being the case we can replace g2 with g1g2 as a generator, and then
we return to the case where only one generator anti-commutes with g.
The same procedure can be utilized when more than two generators anti-
commute with g.
Thus without loss of generality assume that g anti-commutes with a single
generator g1. Note that g has eigenvalues +1 and −1 by the definition of
the Pauli matrices. Since g is Hermitian the Spectral Theorem implies that
the vector space of states under consideration is split in half by g. One half
is the +1 eigenspace, and the other is the −1 eigenspace. This implies that
QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION15
|ψ must lie in one of these spaces, so the result of measuring g for |ψ must
be +1 or −1 and the resulting state will be the projection of |ψ into the
appropriate eigenspace. Mathematically, using the appropriate projection
operators5
:
+1 Measured −→ |ψ+
new =
I + g
√
2
|ψ
-1 Measured −→ |ψ−
new =
I − g
√
2
|ψ
Here the anti-commutativity of g and g1 has been used to determine the
normalization of the new states, after having applied the projection opera-
tors. Now, note that:
P(+1) = | ψ+
new|ψ |2
=
1
2
| ψ|ψ + ψ|g|ψ |
2
= | ψ|ψ + ψ|gg1|ψ |
2
= | ψ|ψ − ψ|g†
1g|ψ |2
= | ψ−
new|ψ |2
= P(−1)
Thus in fact the two measurement outcomes are equally likely. Finally, it is
easy to see that g, g2, ..., gn stabilizes |ψ+
new and −g, g2, ..., gn stabilizes
|ψ−
new . This follows from the commutation of g with the other generators.
The fact that these new stabilizers uniquely specify these new vectors fol-
lows from the following result:
Proposition 10.5 (Nielsen and Chuang): Let S = g1, ..., gN−K be
generated by N − K independent and commuting elements from GN and
such that −I /∈ S. Then VS is a 2K
-dimensional vector space.
We refer the reader to [1] for the proof of this and related results. The
necessity for the generators to be commuting and for −I /∈ S come from
the requirement that VS not be trivial. Nielsen and Chuang show that
VS = {0} if and only if the generators commute and S does not contain
−I.
Given this result, note that the vector spaces stabilized by g1, ...., gn and
±g, g2, ..., gn must have the same dimension if g is independent from
g2, ..., gn. That g is independent from these operators follows by contra-
diction: If g were not then it could be written as some product of powers
of g2, ..., gn. Since the stabilizer originally under consideration g1, ..., gn
5That these operators are the appropriate projection operators follows from the fact that g
has two eigenspaces, with eigenvalues +1 and −1 respectively.
16 DANIEL BULHOSA
stabilizes the non-trivial vector space spanned by |ψ its generators must
commute. In particular this would imply that g and g1 should have com-
muted to begin with, contradicting our original assumption that they do
not. Thus g is independent from g2, ..., gn.
4.4. Error Correction. Now we describe the procedure for constructing a quan-
tum code using the stabilizer formalism. Consider the set GN , which acts on the
space HN . Let S = {g1, ..., gN−K} be a set of independent and commuting elements
of GN . By Proposition 10.5 it follows that the space stabilized by S is 2K
dimen-
sional. Denote this space by VS, then our code will be a map HK −→ VS ⊂ HN .
Now, we must specify the map. Let |x1...xK denote a basis element of HK, so
xi ∈ 0, 1. Given the set S we can find K operators ¯Z1, ..., ¯ZK that commute with and
are independent from the elements of S. We define the logical6
state |x1...xK L ∈ VS
to be the state with stabilizer {g1, ..., gNK
, (−1)x1 ¯Z1, ..., (−1)xK ¯ZK}. Note that this
stabilizer has N elements so by Proposition 10.5 it defines a vector in VS. We map
|x1...xK −→ |x1...xK L.
Motivated by the conjugation relation XZX†
= −Z we define the operators ¯Xj
for 1 ≤ j ≤ K by the condition that ¯Xj
¯Zj
¯X†
j = − ¯Zj and ¯Xj
¯Zk
¯X†
j = ¯Zk for k = j.
Computing the action of Xj on the stabilizer of some logical state shows that it
has the effect of flipping the jth logical qubit.
Now for the error correction: Suppose that Sψ is the stabilizer for a state |ψ ∈
VS. Without loss of generality since errors can be expressed in terms of elements
of GN , let E ∈ GN be an error applied to |ψ . We consider three possibilities:
• E anti-commutes with an element of S: Let g1 be the element with which
E anti-commutes. Let |ψ be an element of the code VS. Then the inner
product of the original state and the erroneous state is:
ψ| E |ψ = ψ| Eg1 |ψ = − ψ| g1E |ψ = − ψ| g†
1E |ψ = − ψ| E |ψ
Here we have use the anti-commutativity relation and the fact that g1 fixes
|ψ . This equation implies that ψ| E |ψ = 0 so the erroneous state is or-
thogonal to the code space. Since this error lies outside the code it can be
detected and corrected.
• E commutes with every element of S, and E ∈ S: Since E ∈ S then E
fixes each element of VS and in particular it fixes the state stabilized by
Sψ. Thus this kind of ”error” has no effect at all!
• E commutes with every element of S, and E /∈ S: When this is the case E
does not fix the elements of VS, but it maps them to different elements of
VS. Thus this kind of error cannot be corrected in general.
6By ”logical” we mean that after the mapping into the code these become our effective com-
putational states, or logic states.
QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION17
Note that if E commutes with every element of S then for g ∈ S we have that
Eg = gE =⇒ EgE†
= g. The subset of E ∈ GN such that this relation holds is
called the normalizer of S, and it is denote by N(S). Note that since each element
of S commutes with every other we have that S ⊂ N(S). The points listed above
constitute the proof of the following theorem:
Theorem: Suppose that E ∈ GN is not an element of N(S) − S. Then the error
E can be corrected.
This completes the principal development of the stabilizer formalism.
5. Concluding Remarks
The stabilizer formalism is an incredibly powerful tool in the field of quantum
error correction. This paper contains sufficient information such that someone
familiar with algebra and information theory should be prepared to understand its
basic applications. For some great examples refer to Chapter 10 Section 5 of [1].
18 DANIEL BULHOSA
References
[1] Michael A. Nielsen and Isaac L. Chuang, 2010: Quantum Computation and Quantum Infor-
mation. Cambridge University Press, 676 pages.
[2] Daniel Gottesman 1997: Stabilizer Codes and Quantum Error Correction. Department of
Physics, Caltech, 114 pages. e-Print: quant-ph/9705052
[3] Roman S. Ingarden, 1976: Quantum Information Theory. Rep. Math. Phys., 10.1, 43-72.
[4] David Deutsch, 1985: Quantum Theory, the Church-Turing Principle and the Universal Quan-
tum Computer. Proc. R. Soc. A, 400, 97-117.
[5] Peter W. Shor, 1994: Polynomial-time algorithms for prime factorization and discrete loga-
rithms on a quantum computer. Proc. 37th Ann. Symp. on the Foundations of Comp. Sci.
(FOCS), 56-65.
[6] Lou K. Grover, 1996: A Fast Quantum Mechanical Algorithm for Database Search. Proc. 28th
Ann. ACM Symp. on the Theory of Comp. (STOC), 212-219.
[7] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, 2009: Quantum Algorithm for Solving
Linear Systems of Equations. Phys. Rev. Lett. 103.15, 150502.
[8] Maria Schuld, Ilya Sinayskiy, and Francesco Petrucionne, 2014: An Introduction to Quantum
Machine Learning. 19 pages. Pre-print: arXiv:1409.3097,
[9] William G. Unruh, 1995: Maintaining Coherence in Quantum Computers. Phys. Rev. A 51.2,
992-997.
[10] Peter W. Shor, 1995: Scheme for Reducing Decoherence in Quantum Computer Memory.
Phys. Rev. A 52.4, R2493-R2496.
[11] Andrew M. Steane, 1996: Error Correcting Codes in Quantum Theory. Phys. Rev. Lett. 77.5,
793-797.
[12] Otto Stern and Walther Gerlach, 1922: Das Magnetische Moment des Silberatoms. Zeitschrift
f¨ur Physik 9, 353-355.
[13] Ramamurti Shankar: Principles of Quantum Mechanics. Plenum Press, New York, 698 pages.
[14] Michael Artin: Algebra. Plenum Press, New York, 698 pages.

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Quantum Computation and the Stabilizer Formalism for Error Correction

  • 1. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION DANIEL BULHOSA Abstract. Computation carried out on machines that exploit the fundamen- tal laws of quantum mechanics have the potential to expediting the solution of certain classes of problems exponentially. Such a machine is known as a quantum computer. Quantum error correction is a field of special interest as it is a prerequisite for the physical implementation of a quantum computer. In this paper we present the stabilizer formalism, which allows for the simple description of error occurrence and correction. A reference for good examples of applications is provided. 1. Introduction Quantum computation promises to exponentially expediting the solution of many difficult computational problems. Early conceptions of quantum information and quantum computing include a seminal paper by Ingarden [3] written in 1976, and a talk by Feynmann in 1981 at MIT describing a basic model for a quantum computer. Deutsch introduced the first model for a universal quantum computer [4]. Shor’s discovery of polynomial time algorithms for solving the discrete logarithm and prime factorization algorithms gave the field of quantum computation its contemporary impetus [5]. The latter is seen as very promising for practical applications since it could be used to break widely-used public-key cryptosystems such as RSA. Grover later introduced an algorithm capable of doing a search in an unsorted database in square root time [6] (as opposed to linear time in a standard computer). Recent research efforts have focused on quantum algorithms that would speed up matrix inversion [7] and machine learning [8] amongst other applications. One of the practical factors that limited the implementation of a quantum com- puter was the lack of effective error correction schemes. An early concern was that the nature of measurement in quantum theory would make it impossible to carry out error correction [9]. In order to correct an error it must be detected first, and researchers were concerned that the process of detecting an error through measure- ment would destroy the information to be corrected. Later it was discovered that it is possible to carry out certain classes of measurements that would allow for error detection without destroying the quantum information [10] [11]. This paper presents a basic introduction to quantum computation. Then it in- troduces the stabilizer sormalism, which utilizes concepts from algebra to create a compact description of quantum error detection and correction. In Section 2 we construct the general mathematical formalism for quantum computation, including Date: May 14, 2015. 1
  • 2. 2 DANIEL BULHOSA the tensor product, states and state spaces, gates, and measurement. In Section 3 we show a few examples of simple quantum information errors and how they are corrected. In Section 4 we introduce the stabilizer formalism and how it can be used to describe errors and their correction. Finally we make some conclusive remarks in Section 5. 2. General Formalism for Quantum Computation In this section we give a basic introduction to the mathematical foundations of quantum computation theory. We first introduce tensor products, a mathematical construct used use two vector spaces to create a new one. We then introduce the basic formalism utilized in quantum computation theory. 2.1. Tensor Products. In order to formally develop the idea of a tensor product we must use the idea of a free vector field. Let S be a set and let K be a field. We define the free vector field FK(S) to be the set of maps f : S −→ K. We think of each such function as describing the sum of symbols of S multiplied by elements of K: f ←→ s∈S f(s)s Note that we have not established any algebraic relations between the elements of s so the summation map is not defined. This is why the formal definition in terms of functions is necessary, but for the sake of intuition we use the summation notation. For example, if S{a, b} and K = R then 1a + πb describes the formal sum f(a) = 1 and f(b) = π. Now, let V and W be vector spaces both over the same field K. We define the tensor product V ⊗ W to be the set FK(V × W) equipped with the following equivalence relations: • Addition in V : (v1, w)+(v2, w) ∼ (v1 +v2, w), where v1, v2 ∈ V and w ∈ W. • Addition in W: (v, w1) + (v, w2) ∼ (v, w1 + w2), where v ∈ V and w1, w2 ∈ W. • Scalar Multiplication: k(v, w) ∼ (kv, w) ∼ (v, kw), where k ∈ K. The operations of addition and multiplication in V ⊗ W are those inherited from FK(V ×W), which are a generalization of the summation and scalar multiplication operations of standard vector fields. We use the common shorthand v ⊗ w for the element (v, w) of V ⊗ W. We can create operators on the tensor space V ⊗W from operators on V and W separately. Let F be an operator on V and let G be an operator on W. We define the tensor product of these two operators, denoted by F ⊗ G, to be the map:
  • 3. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION3 F ⊗ G : V ⊗ W −→ V ⊗ W (F ⊗ G)(v, w) = (Fv, Gw) We use the common shorthand Fv ⊗ Gw for the image of v ⊗ w under F ⊗ G. Often we will deal with extremely large tensor products. For example, consider a system constructed by taking the tensor product of ten identical vector spaces V . Let F and G be operators on V and suppose that we wish to express an operator that applies F to the first and third entries, and G to the tenth entry. In the notation we have presently this operator would be expressed as: F ⊗ I ⊗ F ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ G Here I is the identity operator. This notation is incredibly cumbersome, so we adopt a more convenient one. Let V be a vector space and consider taking its tensor product with itself N times. For 1 ≤ j ≤ N and for any operator F on V we define: Fj = I ⊗ I ⊗ ... ⊗ F at jth slot ⊗... ⊗ I Using this notation we can write the operator previously described as, F1F3G10 which is a significantly more compact description. This follows from the fact that each operator acts on its corresponding slot independently. 2.2. States and State Spaces. The fundamental unit of quantum computation is called the quantum bit, or ”qubit” for short, and it is analogous to the classical bit. Mathematically however, the qubit is very different from the ordinary bit. Whereas the ordinary bit takes on the values 0 and 1, the quantum bit takes on vector values on the Bloch Sphere H: H = {(x, y) ∈ C2 : |x|2 + |y|2 = 1} This definition is motivated by the physical properties of two-state quantum systems1 [1] [13] , which are used to implement qubits physically. If some qubit has value (x, y) we say this tuple is the state of the qubit. Suppose we denote the state (x, y) by the label ψ, then we use the notations:
  • 4. 4 DANIEL BULHOSA |ψ = (x, y) = x y We use the following notation to denote the conjugate transpose of the state ψ: ψ| = |ψ † = (x, y)† = x∗ y∗ Here x∗ denotes the complex conjugate of x. We define the inner product of |ψ and |φ by φ|ψ . This inner product induces the following norm: ψ|ψ = x∗ y∗ x y 1/2 = |x|2 + |y|2 = 1 Note that the set of conjugate transposes forms a vector space, which we denote by H† . We use the tensor product to describe multiple qubit systems. The generaliza- tion of H is a subset of (C2 )⊗N , so we discuss that first. Let (C2 )⊗N denote the tensor product of C2 with itself N times. A generic ele- ment of (C2 )⊗N will be a linear combination of terms of the form |ψ1 ⊗ ... ⊗ |ψN where |ψ1 , ..., |ψN ∈ C2 . Given this we define the set (C2 )⊗N† as the set of linear combinations of elements of the form ψ1| ⊗ ... ⊗ ψN |, where ψ1| , ..., ψN | ∈ (C2 )† and (C2 )† is the set transposes of elements of C2 . The inner product on (C2 )⊗N is induced by that on C2 through the following rule: ( φ1| ⊗ ... ⊗ φN |)(|ψ1 ⊗ ... ⊗ |ψN ) := φ1|ψ1 · · · φN |ψN Here the second line denotes scalar multiplication of the N resulting complex num- bers. Along with the linearity of the inner product this equation entirely specifies its action of any two elements of (C2 )⊗N . Setting φi = ψi for each i yields the formula for the norm. The conjugate transpose is linear and distributes over the tensor product: (|ψ1 ⊗ ... ⊗ |ψN )† = |ψ1 † ⊗ ... ⊗ |ψN † = ψ1| ⊗ ... ⊗ ψN | Having discussed the space C2 we can now define the set of states H: The states of an N qubit system take values in the sets HN defined as: 1One common such system is a single electron. Electrons possess a property called spin, which is can be interpreted as a magnetic dipole generated by the electron.
  • 5. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION5 HN = |ψ ∈ (C2 )⊗N | ψ|ψ = 1 We denote the corresponding subset of (C2 )⊗N† by H† N . Since the states under consideration are elements of (C2 )⊗N for some N we can treat them as vectors and add them together. However, we are only interested in elements of HN so after summing in (C2 )⊗N we must normalize the resulting element. By normalizing, we mean that we must rescale the vector by a complex number so that it will have unit norm. For example, suppose that we want to add the orthonormal states |ψ and |φ together to create a new state. The new state would take the form: |ψ + |φ √ 2 ∈ HN Note the factor of √ 2 dividing the sum. It is easy to check that this normalized state has norm equal to 1 by taking the inner product of it with itself and using the linearity of the inner product and of the adjoint operator: |ψ + |φ √ 2 † |ψ + |φ √ 2 = ψ| + φ| √ 2 |ψ + |φ √ 2 = 1 2 ( ψ|ψ + 2Re( φ|ψ ) + φ|φ ) = 1 2 (1 + 0 + 1) = 1 We call a state that is the normalized sum of other states a superposition of the latter states. Finally, having defined the idea of addition of states we can exploit the fact that HN is a subset of a vector space by utilizing the basis of the latter. Let: |0 = 1 0 , and |1 = 0 1 Then any vector in C2 can be written as a linear combination of this, and thus so can any element of H after appropriate normalization. It is a straightforward result from algebra that if |0 and |1 is a basis for C2 , then the set of vectors, B = {|ψ1 ⊗ ... ⊗ |ψN | |ψi ∈ {|0 , |1 } for each i}
  • 6. 6 DANIEL BULHOSA forms a basis for (Q2 )⊗N . Thus we can describe the elements of H⊗N by taking linear combinations of the elements of B and normalizing appropriately. As a con- vention when using this basis we suppress the tensor notation and consolidate all of the 0’s and 1’s. Thus, for example, we would denote |0 ⊗ |1 ⊗ |1 as |011 . There are special multi-qubit states whose nature partly gives quantum compu- tation its edge over classical computation. These states are called entangled states, which we introduce now for reference. They are defined to be states in which the state of each composite qubit cannot be described separately from the state of the other qubits. Note that: 1 2 (|00 − |01 − |10 − |11 ) = |0 + |1 √ 2 ⊗ |0 − |1 √ 2 This is an example of an unentangled state, as we can describe the composite state as the tensor product of states for the individual qubits. On the other hand the state, |01 + |10 √ 2 does not admit such a factorization so it is entangled. 2.3. Gates. In classical information theory there exists the notion of a logic gate. Mathematically, this is described by the action of some modular arithmetic opera- tion on some set of bits. The notion of a gate is generalized in quantum information theory to the notion of a quantum gate. We begin the discussion by considering H and then generalize to HN . Recall that an operator in a finite dimensional vector space is said to be Her- mitian if F = F† . Here F† denotes the matrix we get by taking the transpose of F and taking the complex conjugate of each entry. Now, it is a straightforward consequence of the definition that the most general Hermitian operator F on C2 takes the form, F = a b∗ b c where a, c are real and b is complex. Write b = x+iy for x, y real, and let w = (a+c)/ 2 and z = (a − c)/2. Then we can re-express the general Hermitian matrix as:
  • 7. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION7 F = w + z x − iy x + iy w − z = w 1 0 0 1 + x 0 1 1 0 + y 0 −i i 0 + z 1 0 0 −1 = wI + xX + yY + zZ The matrices X, Y, and Z are known as the Pauli matrices, and any Hermitian operator in C2 can be expressed as a linear combination of them with the identity. It is easy to check that the square of any Pauli matrix is equal to the identity. Also, note that ZX = iY so we can express Y in terms of the other Pauli matrices. It is useful to note that the commutators and anti-commutators of these matrices obey the following relations: [X, Y ] = 2iZ, [Y, Z] = 2iX, [Z, X] = 2iY {X, Y } = {Y, Z} = {Z, X} = 0 Thus each Pauli matrix either commutes or anti-commutes with the other Pauli matrices. Finally, note that: X |0 = |1 , Y |0 = i |1 , Z |0 = |0 X |1 = |0 , Y |1 = −i |0 , Z |1 = − |1 These relations will prove valuable as we will show any operation on a single quibit can be described by the Pauli matrices. In H we define a quantum operation to be any unitary operator U : H −→ H that can be expressed in the form U = eitF , where F is Hermitian and t ∈ R. This condition is motivated by Schrodinger’s equation, which describes the dynamics of quantum systems. The real variable t in the exponent describes the time the gate has been applied to the qubit. We prove an incredibly useful result: Theorem: Let F be a Hermitian matrix. Then we can express U = eitF in terms of the Pauli matrices and the identity. Proof: Let F = wI + xX + yY + zZ, x = (x, y, z), and ˆx = x/||x||. Also let: ˆx · ˆX = xX + yY + zZ Note that for non-negative integer n:
  • 8. 8 DANIEL BULHOSA (||x||ˆx · ˆX)2k = (xX + yY + zZ) 2 k = x2 X2 + y2 Y 2 + z2 Z2 + xy{X, Y } + yz{Y, Z} + zx{Z, X} k = I (ˆx · ˆX)2k+1 = (xX + yY + zZ) 2k (xX + yY + zZ) = xX + yY + zZ = ˆx · ˆX It is a well known identity that if A and B commute then eA+B = eA eB . Thus we have that: U = eitwI eit||x||ˆx· ˆX = eitw ∞ n=0 (itF)n n! = eitw ∞ k=0 (itF)2k (2k)! + ∞ k=0 (itF)2k+1 (2k + 1)! = eitw ∞ k=0 (it||x||ˆx · ˆX)2k (2k)! + ∞ k=0 (it||x||ˆx · ˆX)2k+1 (2k + 1)! = eitw I ∞ k=0 (−1)k (t||x||)2k (2k)! + i(ˆx · ˆX) ∞ k=0 (−1)k (t||x||)2k+1 (2k + 1)! = eitw I cos(t||x||) + (ˆx · ˆX)i sin(t||x||) We define any quantum operation applied to a qubit intentionally to be a quan- tum gate. This stands in contrast to unintentional quantum operations, which we introduce in the next section. Note that the Pauli matrices are themselves unitary operations, so they describe gates. The following gates, called the Hadamard and phase gates, are important in quantum computation. We state them for reference: T = 1 √ 2 1 1 1 −1 , S = 1 √ 2 1 0 0 i Any operator on a vector space created by taking tensor products of spaces can be described as the tensor product of operators on the tensored spaces. This fol- lows by fixing all but one of the vectors in a tensor product. It follows that any Hermitian operator on HN can be described as the tensor product of the Pauli matrices. In turn, this implies that any quantum operation can be described as the exponential of a tensor product of Pauli matrices. We introduce another important gate for reference. This gate acts on two qubits and it is called the C-NOT gate. It is analogous to the controlled NOT gate in
  • 9. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION9 classical information theory. Denote the C-NOT gate by C, then its action is entirely specified by the following relations: C |0 ⊗ |0 = |0 ⊗ |0 , C |1 ⊗ |0 = |1 ⊗ |0 , C |0 ⊗ |1 = |1 ⊗ |1 , C |1 ⊗ |1 = |0 ⊗ |1 , 2.4. Measurement and State Collapse. Measurement is another major concept in the field of quantum computation that has no analog in classical information the- ory. Classical bits are commonly represented as voltages, and voltages can be en- gineered in such a way that measuring them does not affect their value significantly. In general, measuring a qubit changes its state. It is a postulate quantum me- chanics2 that each measurable physical has associated a Hermitian operator F. When we measure the physical value we retrieve an eigenvalue of this Hermitian operator. The process of measurement changes the state, and the resulting fi- nal state is the eigenvector corresponding to the eigenvalue. Thus the process of measuring can be defined as a probability distribution of eigenvalues fi of F with corresponding eigenvectors |fi : Mψ,F : Eigenvalues(F) −→ [0, 1] Mψ,F (fi) = | fi|ψ |2 For our purposes we can take the set of eigenstates to be finite. It is not obvious that this operation is well defined as it is unclear that the probabilities add up to one. This is the case however when the eigenvalues are distinct. Corollary 8.6.7(a) from [14] guarantees that in vector spaces with inner products like that of HN the eigenvectors of any Hermitian operator form an orthonormal basis.3 This being the case we must have that: 2N i=1 |fi fi|ψ = |ψ =⇒ 2N i=0 | fi|ψ |2 = 2N i=1 ψ|fi fi|ψ = ψ|ψ = 1 The eigenvalue that we measure determines the outcome of the random variable. Thus if we measure fi the new state must be |fi . However, it is apparent that if some of the eigenvalues are equal to each other, or degenerate, then the probabil- ity map is not well defined since the final state itself will not well defined. This problem is solved by generalizing the map in the following way: For j ≤ 2N let f1, ..., fk denote the distinct eigenvalues of F, let mi denote the number of times
  • 10. 10 DANIEL BULHOSA the ith eigenvalue is repeated, and let |fJ i denote the jth orthonormal vector in the subspace of eigenvalue fi. The generalized map is defined as: Mψ,F : Eigenvalues(F) −→ [0, 1] Mψ,F (fi) = mi j=1 | fj i |ψ |2 If the eigenvalue fi is measured the resulting state is the projection of |ψ onto the corresponding eigenspace: |ψ −−−−−−−−→ fj i Measured   mi j=1 |fi fj i |ψ     mi j=1 | fj i |ψ |2   The operator, mi j=1 |fj i fj i | is the appropriate projection operator and the quotient on the right side of the arrow simply guarantees that the final state is normalized to unity. Note that if |ψ is an element of the eigenspace of fi then with probability 1 the eigenvalue fi will be measured and the final state will |ψ itself. 3. Basic Quantum Error Correction Let HK be a system of K qubits. We can map the state of the K-qubit system injectively to a larger system HN , where N ≥ K. We call such a mapping an [N, K] quantum code. Analogously to the classical information theory case, encoding our information allows us to introduce redundancies in it that permit error correction. An example of a simple quantum code is the bit flip code. Let |0 and |1 be our logical states. The bit flip code is the map H1 −→ H3 such that: |0 → |000 , |1 → |111 The states |000 and |111 are our logical states in the code. We define an error to be an unintended change of our information state. This may occur due to noise in the computational hardware or the information channel in which the information 2This axiom is supported by experimental evidence. 3Furthermore the theorem states that the eigenvalues of such an operator must be real. This is important as the values that we retrieve when physically measuring an operator are the eigenvalues of the operator. Physical devices as of yet only measure real numbered quantities.
  • 11. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION11 resides. We can describe any error as the action of a unitary operator on our state, so any error be described in terms of the Pauli matrices and the identity matrix. For example, a bit flip error can be described by the operation of an X operator on one of the qubits: |000 −−−−−→ X2 error |010 Analogously to the classical case however, we can see that we can correct any such single flip error by comparing all three qubit values and flipping the one in dis- agreement. One question naturally arises. Suppose that our erroneous information state is a superposition of states: |010 + |101 √ 2 (1) Intuition would suggest that in order to detect the error we need to measure the states of the individual qubits. However measuring the individual qubits destroys the superposition of states. Suppose for example that we measured the value of the first qubit to be zero. This corresponds to measuring the Z1 operator, which leads to the following transition: |010 + |101 √ 2 −−−−−−−−→ Measurement |010 Superposition is a desirable feature of qubits as they allow for sophisticated computations to be carried out in parallel. Thus is is necessary to come up with measurements that do not destroy the quantum state. For this to be the case these measurements must correspond to operators whose eigenstates are the erroneous states we want to detect. Fortunately in most common cases such measurements exist. In this example measuring the operators Z1Z2, Z2Z3, and Z1Z3 determines ex- actly which bit flip (if any) has occurred. If ZiZj is measured to be +1 qubits i and j have the same value, whereas if −1 is measured they differ. In this example we would measure −1, −1, and +1. This would imply that qubits 1 and 3 agree with each other but qubit 2 disagrees with both of them. Thus qubit 2 was flipped. Note that all of the possible erroneous single qubit flip states, including (1), are eigenstates of these operators. The bit flip error is familiar from classical error correction. There exists an error in quantum computation that does not have a classical counterpart. This is the phase flip error, which is described by the action of the operator Z. For example, consider a phase flip on the second qubit:
  • 12. 12 DANIEL BULHOSA |000 + |111 √ 2 −−−−−→ Z2 error |000 − |111 √ 2 These two vectors are not the same and thus do not represent the same quantum information state. However assuming only a phase flip error has occurred and that no bit flip errors could have occurred we can correct this kind of error as we did in the last example. Measurement of the operator X1X2X3 is sufficient. The central insight in the study of quantum error correction is that any error can be described in terms of bit flip and phase flip errors. This is a consequence of the fact that any generic operation on a set of qubits can be described in terms of the matrices X and Z. Since errors are simply unintended operations on quantum informational states they admit this description. 4. Stabilizer Formalism The stabilizer formalism provides a compact way of describing a large class of error correction codes in quantum information theory. Its key realization realiza- tion in is that we can use operators to uniquely identify certain quantum states. Knowing this, we can describe quantum states in terms of a discrete set of operators rather than as vectors on the Bloch sphere. 4.1. States. To illustrate how a state can be specified using operators, consider the states |000 and |111 . Note that application of the operators Z1Z2 and Z2Z3 returns the same states. By using the matrix and vector representations of these operators and vector respectively simple arithmetic can be used to show that up to a global phase and normalization this is the unique vector satisfying these conditions. Now we introduce the general group whose subsets we will utilize to describe states. Let X, Y , and Z be the Pauli matrices, then we define the group GN to be the set of all N-fold products of the Pauli matrices, allowing for multiplication by ±1 and ±i. This group acts on systems of N qubits. For example, the tensor product −iX1X2Y3Y4Z5 is an element of G5. The subsets that we will utilize are called stabilizers. They are defined as follows: Let S be a subset of Gn, and define VS to be the set of vectors mapped by every element of S to themselves, or fixed by S. Then we say that S is the stabilizer of VS, or that S stabilizes VS. Since the Pauli matrices are linear operators it is easy to show that the set VS is a vector space. Returning to our original example the subgroup, {I1I2I3, Z1Z2, Z2Z3, Z1Z3}(2) stabilizes the two-dimensional vector space spanned by |000 and |111 . Note that X1Z1X2Z2 = −Y1Y2 so since X1X2 and Z1Z2 fix this state so does Y1Y2.
  • 13. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION13 It is often the case that a stabilizer will have many elements, so that specifying it by specifying each element is not practical. However in algebra it is common practice to specify a group in terms of its generators. Let G be some group and let g1, ..., gn be elements of G. We say that the set {g1, ..., gn} generates G if every element of G can be written as a product of elements of this set. We use the nota- tion g1, ..., gn to denote the group generated by the set {g1, ..., gn}. A given group does not necessarily have a unique set of generators. For example, the group (2) is generated by both {Z1Z2, Z2Z3} and {Z1Z2, Z2Z3, Z1Z3}. However the former set is smaller and thus more convenient. In general, for a given group we always find at least one set of generators with minimal size. Such a set is called an independent set of generators, and it is characterized by the condition that removing any generator yields a different group: g1, ..., gl−1, gl+1, ..., gn = g1, ..., gn for each 1 ≤ l ≤ n Independent sets of generators provide the most compact description for a given group, and thus we will utilize them when working with stabilizers. 4.2. Gates. We have devised a way of describing states in terms of sets of stabilizers rather than vectors. Since we use the action of operators to describe changes in states the question naturally arises: How do we describe the effect of an operator in terms of stabilizers? The answer is in fact quite intuitive. Note that U is a unitary operator and some operator W fixes |ψ then the operator UWU† fixes U |ψ for: (UWU† )U |ψ = (UW)(U† U) |ψ = UW |ψ = U |ψ This makes it clear that if S is the stabilizer for the state |ψ then, USU† := {UWU† | W ∈ S} is the stabilizer for W. Thus the action of a unitary operation U on a state repre- sented by S is conjugation by U.4 In the stabilizer formalism the action of the basic quantum gates introduced in Section 2.3 of the paper is simple. For example, the action of the Hadamard gate on the Pauli matrices is: HXH† = Z, HY H† = −Y, HZH† = X Though not all possible gates can be described as combinations of these basic gates, most conventional quantum error corrections schemes can be described in terms of 4If W and U are operators we say that UWU† is the conjugate of W with respect to U. If S is a set of operators, we call USU† the conjugate set to S with respect to U.
  • 14. 14 DANIEL BULHOSA them. Furthermore it can be shown that any unitary operation mapping elements of Gn to Gn can be described in terms of the Hadamard, phase, and C-NOT gates [1]. Thus within the framework of the stabilizer formalism these basic gates allow for a simple but complete description of the error correction process. 4.3. Measurement. Just as the process of measurement changes the state vector of a system it also changes its stabilizer. The change that the stabilizer undergoes depends on the relation of the observable g being measured to the generators of the stabilizer of the state under consideration. Suppose that the state under consideration |ψ has a stabilizer S = g1, ..., gn , where this set of generators is independent. Without loss of generality we can assume that g is an element of GN with no multiplicative factor of ±i or −1 in front of it. This assumption can be made as multiplication by an overall factor does not change the eigenvalues and eigenvectors of g. This being the set-up of our measurement problem, we have two possibilities: • The observable g commutes with every element of S. In this case we have for each gi ∈ S that gi(g |ψ ) = ggi |ψ = g |ψ . Thus g |ψ is stabilized by S and thus it must be parallel to |ψ . Furthermore, since g is a tensor product of Pauli matrices we must have that g2 = I. These two facts com- bined imply that g |ψ = ± |ψ . Thus we have shown that |ψ is an eigenvector of g in this case. Recall that measurement of an operator for a state that is in an eigenspace of the former leaves the latter unchanged. Thus the final state has an identical stabilizer. • The observable g anti-commutes with a single element of S. Note that this is entirely general: For Example, g does not commute with two elements g1 and g2 of S, then g does commute with g1g2: [g1g2, g] = g1g2g − gg1g2 = −g1gg2 − gg1g2 = g(g1g2 − g1g2) = 0 This being the case we can replace g2 with g1g2 as a generator, and then we return to the case where only one generator anti-commutes with g. The same procedure can be utilized when more than two generators anti- commute with g. Thus without loss of generality assume that g anti-commutes with a single generator g1. Note that g has eigenvalues +1 and −1 by the definition of the Pauli matrices. Since g is Hermitian the Spectral Theorem implies that the vector space of states under consideration is split in half by g. One half is the +1 eigenspace, and the other is the −1 eigenspace. This implies that
  • 15. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION15 |ψ must lie in one of these spaces, so the result of measuring g for |ψ must be +1 or −1 and the resulting state will be the projection of |ψ into the appropriate eigenspace. Mathematically, using the appropriate projection operators5 : +1 Measured −→ |ψ+ new = I + g √ 2 |ψ -1 Measured −→ |ψ− new = I − g √ 2 |ψ Here the anti-commutativity of g and g1 has been used to determine the normalization of the new states, after having applied the projection opera- tors. Now, note that: P(+1) = | ψ+ new|ψ |2 = 1 2 | ψ|ψ + ψ|g|ψ | 2 = | ψ|ψ + ψ|gg1|ψ | 2 = | ψ|ψ − ψ|g† 1g|ψ |2 = | ψ− new|ψ |2 = P(−1) Thus in fact the two measurement outcomes are equally likely. Finally, it is easy to see that g, g2, ..., gn stabilizes |ψ+ new and −g, g2, ..., gn stabilizes |ψ− new . This follows from the commutation of g with the other generators. The fact that these new stabilizers uniquely specify these new vectors fol- lows from the following result: Proposition 10.5 (Nielsen and Chuang): Let S = g1, ..., gN−K be generated by N − K independent and commuting elements from GN and such that −I /∈ S. Then VS is a 2K -dimensional vector space. We refer the reader to [1] for the proof of this and related results. The necessity for the generators to be commuting and for −I /∈ S come from the requirement that VS not be trivial. Nielsen and Chuang show that VS = {0} if and only if the generators commute and S does not contain −I. Given this result, note that the vector spaces stabilized by g1, ...., gn and ±g, g2, ..., gn must have the same dimension if g is independent from g2, ..., gn. That g is independent from these operators follows by contra- diction: If g were not then it could be written as some product of powers of g2, ..., gn. Since the stabilizer originally under consideration g1, ..., gn 5That these operators are the appropriate projection operators follows from the fact that g has two eigenspaces, with eigenvalues +1 and −1 respectively.
  • 16. 16 DANIEL BULHOSA stabilizes the non-trivial vector space spanned by |ψ its generators must commute. In particular this would imply that g and g1 should have com- muted to begin with, contradicting our original assumption that they do not. Thus g is independent from g2, ..., gn. 4.4. Error Correction. Now we describe the procedure for constructing a quan- tum code using the stabilizer formalism. Consider the set GN , which acts on the space HN . Let S = {g1, ..., gN−K} be a set of independent and commuting elements of GN . By Proposition 10.5 it follows that the space stabilized by S is 2K dimen- sional. Denote this space by VS, then our code will be a map HK −→ VS ⊂ HN . Now, we must specify the map. Let |x1...xK denote a basis element of HK, so xi ∈ 0, 1. Given the set S we can find K operators ¯Z1, ..., ¯ZK that commute with and are independent from the elements of S. We define the logical6 state |x1...xK L ∈ VS to be the state with stabilizer {g1, ..., gNK , (−1)x1 ¯Z1, ..., (−1)xK ¯ZK}. Note that this stabilizer has N elements so by Proposition 10.5 it defines a vector in VS. We map |x1...xK −→ |x1...xK L. Motivated by the conjugation relation XZX† = −Z we define the operators ¯Xj for 1 ≤ j ≤ K by the condition that ¯Xj ¯Zj ¯X† j = − ¯Zj and ¯Xj ¯Zk ¯X† j = ¯Zk for k = j. Computing the action of Xj on the stabilizer of some logical state shows that it has the effect of flipping the jth logical qubit. Now for the error correction: Suppose that Sψ is the stabilizer for a state |ψ ∈ VS. Without loss of generality since errors can be expressed in terms of elements of GN , let E ∈ GN be an error applied to |ψ . We consider three possibilities: • E anti-commutes with an element of S: Let g1 be the element with which E anti-commutes. Let |ψ be an element of the code VS. Then the inner product of the original state and the erroneous state is: ψ| E |ψ = ψ| Eg1 |ψ = − ψ| g1E |ψ = − ψ| g† 1E |ψ = − ψ| E |ψ Here we have use the anti-commutativity relation and the fact that g1 fixes |ψ . This equation implies that ψ| E |ψ = 0 so the erroneous state is or- thogonal to the code space. Since this error lies outside the code it can be detected and corrected. • E commutes with every element of S, and E ∈ S: Since E ∈ S then E fixes each element of VS and in particular it fixes the state stabilized by Sψ. Thus this kind of ”error” has no effect at all! • E commutes with every element of S, and E /∈ S: When this is the case E does not fix the elements of VS, but it maps them to different elements of VS. Thus this kind of error cannot be corrected in general. 6By ”logical” we mean that after the mapping into the code these become our effective com- putational states, or logic states.
  • 17. QUANTUM COMPUTATION AND THE STABILIZER FORMALISM FOR ERROR CORRECTION17 Note that if E commutes with every element of S then for g ∈ S we have that Eg = gE =⇒ EgE† = g. The subset of E ∈ GN such that this relation holds is called the normalizer of S, and it is denote by N(S). Note that since each element of S commutes with every other we have that S ⊂ N(S). The points listed above constitute the proof of the following theorem: Theorem: Suppose that E ∈ GN is not an element of N(S) − S. Then the error E can be corrected. This completes the principal development of the stabilizer formalism. 5. Concluding Remarks The stabilizer formalism is an incredibly powerful tool in the field of quantum error correction. This paper contains sufficient information such that someone familiar with algebra and information theory should be prepared to understand its basic applications. For some great examples refer to Chapter 10 Section 5 of [1].
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