Quantum error correction with stabilizer codes

In our previous discussion of quantum error correction, we have assumed that quantum gates can act on any two physical qubits. In reality, however, this is not true – only nearby qubits and interact, and our error correction needs to take the geometric arrangements of the qubits into account. The link between these geometric constraints and the theory of quantum error correction is the stabilizer formalism described first in [2].

This post will be a bit more formal, but is a necessary preparation to be able to define and understand the surface code and other topological codes. To introduce and motivate the formalism, let us for a moment come back to the simple example studied before – the three-bit code. In this code, the logical states were

|0 \rangle_L = |000 \rangle

and

|1 \rangle_L = |111 \rangle

These logical states span our two-dimensional code space and describe one logical qubit. A single bit flip error will flip one of the three involved qubits. Suppose, for instance, that a bit flip occurs on the first or second qubit. This error will modify our logical states as follows (the first two lines represent a bit flip on the first qubit, the next two lines a bit flip on the second qubit

|0 \rangle_L \mapsto |100 \rangle

|1 \rangle_L \mapsto |011 \rangle

|0 \rangle_L \mapsto |010 \rangle

|1 \rangle_L \mapsto |101 \rangle

Now consider the operator

M_1 = \sigma_z^1 \sigma_z^2

where the upper index on a Pauli matrix indicates the qubit on which the operator acts. The error-free logical states are both eigenstates of this operator with eigenvalue +1. The states that result out of a bit-flip error are also eigenstates of this operator, but with eigenvalue -1. Thus, a measurement of this operator will tell us whether a bit flip on the first or second qubit has occurred. Similarly, to also detect a bit flip on the third qubit, we need a measurement for a second operator, namely

M_2 = \sigma_z^1 \sigma_z^3

Let us now write down a few properties of these operators. First, they are hermitian and therefore correspond to measurements. Second, the logical states are eigenvectors of these operators with eigenvalues +1 and the code space is exactly the subspace of the three-qubit state space that is fixed by M1 and M2. We can think of these operators as defining linear constraints that together define the code space, as indicated below.

CodeSpaceStabilizer.png

Moreover, if a bit flip error occurs, the resulting state will again be an eigenvalue, but for at least one of the operators the new eigenvalue will be -1. Thus, errors correspond to violated constraints. Finally, the two operators commute and can therefore be measured simultaneously.

These properties allow us to express the three bit code entirely in terms of the operators M1 and M2. The code space is the subspace which is left invariant by these operators. The syndrome measurement can be done by measuring both operators simultaneously, which is possible because they commute. Finally, all involved states – code space and states after bit flip errors – are eigenstates and therefore the measurement process does not collapse a superposition of these states and can therefore be executed without interfering with the actual quantum algorithm that we want to run.

This correspondence between sets of operators – the Mi in our case – and quantum error correction is the core of the stabilizer formalism. In a more general setting, we are looking at the Hilbert space of dimension 2n spanned by n physical qubits. Products of n Pauli matrices (where we include the identity matrix) are acting on this Hilbert space and form a group {\mathcal G}_n called the Pauli group. Put differently, the Pauli group consists of those linear operators on the Hilbert space that can be written as a product

g = \mu (\sigma_x^1)^{a_1} (\sigma_y^1)^{b_1} (\sigma_z^1)^{c_1}\cdots (\sigma_x^n)^{a_n}(\sigma_y^n)^{b_n} (\sigma_z^n)^{c_n}

with coefficients a_i, b_ic, c_i and an overall phase factor \mu \in \{ \pm 1, \pm i \} (we can even assume that all the b_i are equal to one as \sigma_x \sigma_z is a multiple of \sigma_y). Any two elements of the Pauli group either commute or anti-commute. Within this group, we now consider a finite set \{ M_i \} of commuting elements and the subgroup S of the Pauli group generated by this set. This group (which is abelian as it is generated by mutually commuting elements) is called the stabilizer group. To this group, we can associate the subspace that consists of all vectors that are fixed by all elements of the group, i.e. the space

T_S = \{ |\psi \rangle \, \text{such that} \, s|\psi \rangle = |\psi \rangle  \forall s \in S \}

This space will be the code space of our code. If the group S is generated by l independent generators, the dimension of the code space can be seen to be 2n-l.

In the example of the three bit code, we had n=3 and l=2, which gives us a two-dimensional code space, corresponding to one logical qubit (if you try work out the details, you will see that in order for this to be true, we have to assume that -1 \notin S – you might want to take a look at my notes or the chapters on quantum error correction in [1] for the mathematical details).

Which errors is our code able to detect? Suppose that E is an error operator that is itself in the Pauli group. We know that any two elements of the Pauli group either commute or anti-commute. Let us assume that E does in fact anti-commute with some element s of S. If |\psi \rangle is a state in the code space, we can then calculate

s E |\psi \rangle = - E s |\psi \rangle  = - E |\psi \rangle

Therefore the state E |\psi \rangle is now in the -1 eigenspace of s. Thus if we measure all elements of S, the outcome -1 for one of the measurements will tell us that an error has occurred. A similar argument shows that if the product of any two errors anti-commutes with at least one element of S, then we can also correct the error based on only the measurement results of elements in S. Mathematically speaking, the set of all elements of the Pauli group that commute with all elements of S is the centralizer of S, which in this case turns out to be equal to the normalizer N(S) of S, and a set \{E_\alpha \} of errors can be detected and corrected if and only if

E_\alpha E_\beta \in S \cup (\mathcal{G} - N(S))

for any two error operators E_\alpha, E_\beta. So the elements of the Pauli group that are neither in S nor in N(S) correspond to correctable errors.

What do the elements of the normalizer correspond to? Suppose that n is an element in the normalizer and therefore commutes with all elements of S. Then, for any element s in S and any element |\psi \rangle of the code space, we have

s n |\psi \rangle = n s |\psi \rangle = n |\psi \rangle

Consequently, the element n |\psi \rangle is again fixed by all elements of S and therefore in the code space TS. Thus the elements of N(S) generate automorphisms of the code space, i.e. logical operations. It is one of the strengths of the stabilizer formalism that once we have the stabilizer group S, we can not only derive the code space and the correctable errors, but can also use group theoretic considerations to describe logical operations – this will become clearer as we study surface codes in a later post.

For later reference, let us briefly summarize what we have found. Given l independent generators si of the stabilizer S, we can define a code space as the set of all vectors that are fixed by the si. This subspace will encode n – l logical qubits in n physical qubits. To detect an error, we measure all the si. Each measurement will give us minus one or plus one. Any occurrence of minus one indicates an error (this is why the combined result of all measurements is usually called the error syndrome) and will also tell us which operation we have to apply to correct the error. To implement logical quantum gates on our code space, we can apply elements in the normalizer N(S) that map the code space into itself. Thus we have expressed error detection, error correction and logical operations entirely in terms of the stabilizer group and the language of group theory.

But how do we find useful stabilizer codes? One approach is given by the check matrix formalism, which allows us to express stabilizer codes in terms matrices and metrics, i.e. in terms of linear algebra. This approach is generalizing the parity check matrix from the classical theory of error correction. A second source of stabilizers, however, comes from a more surprising direction – algebraic topology. In fact, given a surface described in terms of a cell complex, we can associate elements of the Pauli group with every cycle and every co-cycle. For certain choices of cycles and cocycles, this will give us abelian subgroups of the Pauli group which in turn create error correction codes. This is how geometrical constraints in the actual implementation enter the scene and leads to a class of codes known as surface codes and toric codes that we will start to study in my next post.

1. E. Rieffel, W. Polak, Quantum computing – a gentle introduction, MIT Press, Cambridge 2011
2. D. Gottesman, Stabilizer Codes and Quantum Error Correction, Caltech Ph.D. Thesis, available as arXiv:quant-ph/9705052
3. J. Kempe, Approaches to quantum error correction, S\’eminaire Poincar\’e 1 (2005), pp. 65–93, available as arXiv:quant-ph/0612185

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