Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes
We discuss and review several thermodynamic criteria that have been introduced to characterize the thermal stability of a self-correcting quantum memory. We first examine the use of symmetry-breaking fields in analyzing the properties of self-correcting quantum memories in the thermodynamic limit: we show that the thermal expectation values of all logical operators vanish for any stabilizer and any subsystem code in any spatial dimension. On the positive side, we generalize the results in [R. Alicki et al., arXiv:0811.0033.] to obtain a general upper bound on the relaxation rate of a quantum memory at nonzero temperature, assuming that the quantum memory interacts via a Markovian master equation with a thermal bath. This upper bound is applicable to quantum memories based on either stabilizer or subsystem codes.
Thermal fluctuations pose a serious problem for reliable, passive, information storage since any open system eventually reaches a thermal equilibrium state in which all encoded information is lost. Fortunately, it has been shown that quantum information can be reliably stored for arbitrary long times in, say, a 2D quantum memory  by means of active error correction and entropy removal. However, the implementation of active error correction implying extensive and fast classical input/output to the quantum memory poses a serious (but hopefully not insurmountable) experimental challenge.
The central idea behind self-correcting classical or quantum memories is to do without active error-correction and prevent thermalization and build-up of entropy by the presence of macroscopic “energy barriers” separating encoded states.
The idea of such self-correcting quantum memory was first introduced in  and can be viewed as an extension of the ideas of topological protection developed by Kitaev . In  it was argued that the 2D surface or toric code (2D Kitaev model) would not be a self-correcting memory, but a 4D surface code (4D Kitaev model) generalization was presented which would be thermally stable. Unfortunately, three spatial dimensions is all the room that the natural world seems to provide.
It is thus of interest to (1) either come up with models for self-correcting quantum memories in 3 or fewer dimensions, or (2) show that low-dimensional quantum physics does not allow for such passive stability. The latter possibility would provide evidence that genuine quantum phases of nature, such as topological phases, would be confined to the domain of finite systems and low temperatures: in the thermodynamic limit thermal fluctuations would destroy the quantum order at any nonzero temperature. Such a no-go possibility would also lend support to the thought that macroscopic quantum states suffer from intrinsic decoherence (see  for thoughts in this direction). In this sense we believe that the question of thermal stability of a passive quantum memory is one of fundamental interest.
In fact, the thermal stability question can also be viewed as a question about the nature of the excitations of the quantum memory model. For 2D topological models these excitations are point-like pairs of anyons. If we paraphrase the macroscopic energy barrier requirement of  in terms of the nature of excitations, it relates to a condition that the elementary excitations are extended objects; they are the boundary of a two or higher-dimensional surface.
In  the subject of self-correcting quantum memories was brought to the fore. Bacon introduced two models, now called the 2D Bacon-Shor code or quantum compass model, and the 3D Bacon-Shor code, which are examples of quantum subsystem codes. The 3D Bacon-Shor model may or may not be an example of a self-correcting quantum memory; it is an open question how to analyze its thermal stability.
The analysis of the thermal stability of a quantum stabilizer or quantum subsystem code model in a thermodynamic sense is the subject of this paper. Let us discuss some of the literature on this subject.
Necessary criteria for thermal stability of a quantum memory were formulated in  (see also ) in terms of a macroscopic distance of the underlying quantum code (i.e. zero temperature topological order) and the presence of macroscopic energy barriers. It was shown in that paper that all 1D and 2D local stabilizer codes fail to meet these criteria. The advantage of this approach is that it allows for very general no-go results. A disadvantage is that it does not make contact with any operational or thermodynamic expression of thermal stability. In particular, to prove positive results on particular quantum memory models, it is necessary to more thoroughly analyze the thermodynamics of an open quantum memory. The intuition that underlies the idea of a self-correcting quantum memory is that errors of increasing weight should map encoded states onto excited states with increasingly higher energy. If the quantum code has a macroscopic distance which scales with system-size, then high-weight errors will have to happen in order to map one encoded state onto another. But such high-weight errors will, if the memory is self-correcting, correspond to high-energy states, hence there would be (macroscopic) energy barriers between different encoded states. In the second part of our paper, Section 7 we will indeed see that the energy associated with high-weight errors corresponding to so-called bad syndromes, will play a crucial role in bounding the quantum memory relaxation rate.
Specific results ruling out the existence of finite temperature topological order for e.g. 2D toric code, were obtained in [7, 8, 9], using in  an interesting finite-temperature extension of the topological entanglement entropy. Remarkably, these limitations can be overcome by including repulsive long-range interactions with bounded strength. Such extensions of the 2D toric code were proposed in  and are characterized by a diverging relaxation time in the thermodynamic limit. Since the requirement of a macroscopic energy barrier between logical states [4, 6] is violated in these models, the increase of the lifetime with the system size is only polynomial. However, the scaling power is very sensitive to the physical features of the thermal bath and becomes especially favorable for super-ohmic reservoirs. Such properties needed to be established in  by the explicit analysis of the non-equilibrium time evolution, instead of being addressed via a suitable equilibrium quantity as in the present work.
In  a thermodynamic criterion was presented for the existence of topological order at finite temperature. There, it was discussed whether the thermal expectation value of logical qubit operators could serve as a stability criterion for a quantum memory against thermal fluctuations. Specifically, following the reasoning used in the discussion of spontaneous symmetry breaking, a small perturbation (external field) is applied to the system which breaks explicitly the symmetry of the Hamiltonian and the state of the system. Then, the thermodynamic limit is taken before the external field is taken to zero. If the expectation values of the logical operators vanish in this order of limits, then, according to the argument given in Ref. , the information in the quantum memory will be lost after a finite, size-independent relaxation time at any finite temperature. This concept was demonstrated explicitly for the Kitaev model in 2D and for some generalizations of it to higher dimensions .
In this paper we will discuss and analyze this criterion. By making use of elementary arguments we show that the same line of reasoning as in Ref.  allows one to go well beyond these results: in particular, zero thermal averages for the logical operators are obtained not only independently of any microscopic details of the code, be it a stabilizer code or a subsystem code, but also in any spatial dimension. We will discuss the root cause of these problems and discuss possible ways to extend the traditional analysis of spontaneous symmetry breaking to detecting a finite temperature quantum order.
The analysis of thermal stability of a quantum memory within the formalism of the thermodynamics of open quantum systems was seriously undertaken in a series of papers by Alicki and co-workers [11, 12, 13]. In  it was demonstrated that for the 2D surface code model weakly coupled to a Markovian environment, the relaxation rate of any logical state is bounded from below by a constant independent of system size . This result implies that increasing the system size does not increase the lifetime (stability) of the memory, but that the relaxation rate is an intrinsic feature of the model. In  the authors considered the 4D Kitaev model and explicitly proved that the relaxation times were exponentially increasing with system size, hence confirming the anticipated thermal stability in the thermodynamic limit.
In the second part of our paper (Sections 6,7) we will present a formal analysis of the thermal stability of a quantum memory based on subsystem (stabilizer) codes . The difficulty for Hamiltonian models based on subsystem codes (see discussions in ) is that the Hamiltonian is a sum of non-commuting terms, hence spectral information for such systems is not readily available analytically. Using some of the ideas developed in [9, 11, 12, 13] we will construct a simple observable whose expectation value on the thermal Gibbs state provides an upper bound on the relaxation rate thus determining how long quantum information can be stored in a given system. Our formalism will be general enough to cover both stabilizer as well as subsystem codes. In addition, we can use this formalism to provide a simple bound on the memory relaxation time of stabilizer of subsystem code which is not self-correcting, but is ‘protected by a gap’. In Section 8 we prove that the memory relaxation time scales as where is the system size, is the inverse temperature, and is the spectral gap of the memory Hamiltonian. For sufficiently small temperature, e. g. logarithmically scaling with the system size, such models may still be of practical interest.
At the end of the paper we will show that the bound on the relaxation rate only depends on an induced temperature-dependent distribution associated with the Abelian stabilizer group of the subsystem gauge group.
2 Stabilizer and subsystem codes
We assume that the system chosen as the storage medium is represented by an -qubit Hilbert space . Let be the Pauli group on qubits generated by single-qubit Pauli operators and the phase factors , . We envision that the quantum data is stored in the degenerate ground states of a quantum Hamiltonian acting on the physical qubits. The Hamiltonian will be associated with a quantum stabilizer or subsystem code.
A subsystem code is determined by its gauge group which can be an arbitrary subgroup of . The set of Pauli operators that commute with all elements of is called the centralizer of and is denoted as . The Abelian group is called the stabilizer group of . If is Abelian, then obviously up to phase factors and we call a stabilizer code. To preclude from containing non-trivial phase factors one usually adds a requirement in the case of stabilizer codes. If is non-Abelian, we refer to as a subsystem code.
Logical operators of a stabilizer code are elements of which are not in . One can always choose a set of logical Pauli operators obeying the usual Pauli commutation relations: and . Note that .
The code space of a stabilizer code is defined as the common dimensional eigenspace of . It can also be viewed as the ground space of a Hamiltonian acting on qubits:
Here the are some real negative coupling constants and the operators form an (over)complete set of generators of . Note that the definition of a logical operator is not unique, since we can multiply any logical operator by an element in which acts trivially on any state in the code space/ground space. Note that the logical operators are symmetry operations of since they commute with all elements thus each energy level of has a degeneracy . Therefore, the choice of the ground space as coding space, instead of any of the higher energy levels, is somehow arbitrary and other forms of encoding might be more useful. An interesting example is the thermal state encoding which will be described in Section 6.1.
Bare logical operators of a subsystem code are elements of the centralizer which are not in . One can always choose a set of bare logical Pauli operators obeying the usual Pauli commutation relations. Note that . We can multiply such bare logical operators by elements in to get so-called dressed logical operators, which act on the gauge qubits, in addition to the logical qubits. With the group and its local generators we can associate a Hamiltonian
where are some real coefficients. Since any commutes with all the bare logical operators , it follows that commutes with . In addition, commutes with all elements in the Abelian stabilizer group of , hence is block-diagonal in sectors labeled by the quantum numbers (syndromes) of this stabilizer group . Typically, ground states of are confined to a single syndrome sector.
For simplicity we will assume in the remainder of this paper that a single qubit is encoded in the quantum memory, i.e. .
3 Thermal fragility?
To get started, let us consider the thermal fragility criterion introduced in  and apply this to general stabilizer code Hamiltonians, Eq. (1). As in Ref. , we introduce where the additional perturbation is a symmetry-breaking field, designed to produce a finite expectation value of the logical operators for the encoded qubit. Here .
For simplicity, we will consider a perturbation along the -direction (this can always be assumed by a suitable choice of the logical operators), i. e. and .
We can write the degenerate eigenvectors of with energy as where is the eigenvalue of (the and the operator can be diagonalized simultaneously).
Clearly, only acts on the quantum number (the error syndrome, see Section 4) of the eigenfunctions , while the logical operators, and in particular the perturbation , only acts on the quantum numbers. As a consequence, the canonical partition function at temperature factorizes
where we used , independent of .
As was shown in , we immediately see that the average value is independent of the unperturbed Hamiltonian , and only reflects the finite degeneracy of the energy levels
The expectation value in Eq. (4) evidently goes to zero at small . It is clear that Eq. (4), being independent of the form of , holds also if the unperturbed Hamiltonian refers to a macroscopic system. Therefore, this procedure yields vanishing averages also after the thermodynamic limit is taken. If the Hamiltonian involves physical qubits, we get
Given that this argument is independent of dimensionality, and thus also holds for the 4D Kitaev model which is believed to be thermally stable, the result suggests that the symmetry-breaking field used in is not strong enough to bias the thermal state towards having a non-zero logical operator expectation value.
Proof based on the Bogoliubov inequality. We consider next an alternative approach based on the Bogoliubov inequality [15, 16] and show that we reach the same conclusion as before. This method can then be applied to subsystem codes (see Section 3.1). We start from the well-known Bogoliubov inequality [15, 16]
where are two arbitrary operators and is the system Hamiltonian, with the assumption that all expectation values exist. Here we use the convention . We then set and . Clearly, and the right-hand-side of the Bogoliubov inequality (6) gives . Therefore we get
where the operator commutes with . Using the commutation relations for the logical operators, we obtain . For strictly positive (otherwise we are done) we can divide by , and then, by taking the thermodynamic limit on both sides of the resulting inequality, we eventually get
At any finite temperature, we thus obtain that the thermal expectation value of the logical operator vanishes when .
3.1 Subsystem codes
Let us use the Bogoliubov inequality to argue about the thermal fragility criterion for subsystem codes. The bare logical operators of the encoded qubit commute with all , hence with in Eq. (2). We can choose the symmetry-breaking Hamiltonian as
for some choice of which dresses the bare logical operator . Let us thus consider the thermal expectation value of where does not need to be the same as .
We use the Bogoliubov inequality, with and . Since commutes with and (valid for every Pauli operator), one obtains . This gives
and since commutes with and , we can easily compute the left-hand side, to obtain
We now notice that is a Pauli operator and thus has eigenvalues . Hence, the thermal expectation value on the left side is always less then 1 independently of any details of , which gives
for any choice of and . Therefore, the same considerations valid for the stabilizer codes can be repeated in this case and we again conclude that the thermal expectation value of any logical operator vanishes at any finite temperature for vanishing field .
4 Error correction
Let us pause for a moment and discuss our somewhat naive-looking approach. It seems that there are at least two issues at stake here. Let us assume that by choosing the right symmetry-breaking field, we are able to concentrate the weight of around a logical . Consider this eigenstate of the logical operator and a state with a single qubit error, , such that anti-commutes with . Obviously, if at equilibrium the system is in a statistical mixture of and with equal probability, one has . However, the information in the memory is still preserved as long as we correct for errors such as when we determine what logical state has been stored. For a generic stabilizer code, the probability of the states is small at low temperature (below the gap), but the statistical weight of all correctable errors might be very large in the thermodynamic limit. Therefore, does not represent a meaningful stable order parameter for this problem; the value of has to be modified depending on the error syndrome. For this reason, the authors of Ref.  consider so-called error-corrected logical operators 111In  these are called dressed logical operators, but we prefer to reserve the notion of ‘dressing’ for the multiplication of bare logical operators with elements of the gauge group .. Let us properly define these for stabilizer and subsystem codes.
For stabilizer codes, error correction consists of measuring the eigenvalues of the stabilizer generators; these sets of eigenvalues form the error syndrome. The error syndrome is used as input to a classical decoding algorithm which determines which errors have most likely taken place. For subsystem codes, error correction may proceed by measuring the eigenvalues of the local generators . Since the operators do not commute, these eigenvalues cannot be simultaneously measured, nonetheless these (random) values of the generators of will fix the eigenvalues of the stabilizer group . These eigenvalues of the stabilizer group again form the error syndrome.
More precisely, any error determines a syndrome such that
We can assume that there is some deterministic decoding algorithm which assigns a correcting Pauli operator to every syndrome . An error is correctable iff coincides with up to a gauge operator, that is, .
We can define a subspace projector associated with every syndrome (quantum number) . Let be the projector onto the -invariant code space in which for all . (By abuse of notations let us assume from now on that .) For any syndrome we can define where is any error with syndrome (note that the projector does not depend on the choice of such ). Clearly . We define an error-correcting transformation for observables on as
Note that , so the adjoint transformation acting on states is a trace-preserving completely-positive (TPCP) map. Following  we can define the error-corrected logical operators as
for a pair of bare anti-commuting logical operators . Note that are not necessarily Pauli operators. However, it is not hard to show that the error-corrected logical operators obey the relations and . We can understand this by defining coefficients such that
Any syndrome projector belongs to the algebra generated by and thus commutes with . It follows that
The commutation relations for follow directly from Eq. (16). Note also that the error-corrected logical operators commute with all elements in .
We can immediately check whether the use of error-corrected logical operators would change the analysis of the thermal expectation values. As observable, we choose, say, for some whereas for the symmetry-breaking field we choose some . Using the properties of stated above, we can repeat the proof of the previous subsection to obtain again a vanishing expectation value
5 Analogy with the 2D Ising model: choice of symmetry-breaking field
We emphasize that the conclusions above are valid for arbitrary dimensions of any stabilizer or subsystem code. Although (or since) the argument is so universal it also appears to be exceedingly oversimplified. In the previous section, we have discussed the necessity to choose a stable logical observable which includes the process of error correction. Let us now more closely examine the choice for the symmetry-breaking field.
Although the thermal fragility criterion is patterned along the lines of standard symmetry-breaking arguments, it is only so on a formal level. It is instructive to compare the argument of Ref.  with the standard example of spontaneous symmetry-breaking in the 2D Ising model  (see e. g. ):
where label the 2D sites of the full lattice , the first sum is over pairs of nearest neighbor sites, and an external magnetic field is included. For the 2D Ising model one obtains at low temperature
at every lattice site , where the expectation value above is taken with respect to the Hamiltonian (18). This appearance of a symmetry-breaking order should be contrasted with the lack of such order in the 1D Ising model which has .
Notice that, although the 2D Ising model does not display topological order, it does define a proper stabilizer code with logical operators and , where is a fixed (arbitrary) site in the lattice222Of course, the other two expectation values and are vanishing in the appropriate thermodynamic limit. This stabilizer code does not provide a good quantum memory since the distance of the code is 1 independent of lattice size..
The arguments discussed in the previous sections consider a perturbation which leads to and does not show that the value of the -polarization is robust. In fact, the field only acts on a single site, whereas in the standard case the symmetry breaking field acts on all sites of the lattice simultaneously, see Eq. (18). The reason for the failure of the stability criterion appears thus to be that the chosen symmetry-breaking perturbation is not extensive. Although for topological memories the support of a logical operator (i.e., the number of physical spins on which the operator acts nontrivially) becomes larger with the size of the system, the perturbation is bounded in norm by and becomes irrelevant in the thermodynamic limit.
The analogy with the 2D Ising model suggests that the symmetry-breaking field should be chosen as a sum over different incarnations of a logical operator, i.e. we can multiply a logical by elements of the stabilizer code and obtain an extensive operator. It may be possible to salvage this symmetry-breaking route to getting a quantum order parameter, but of course any construction should ultimately be motivated operationally. This is the reason that we now switch to explicitly deriving a memory relaxation rate.
6 Relaxation rate for general quantum memory Hamiltonians
The goal of this section is provide a criterion for thermal stability for a large class of quantum systems that can be described by subsystem codes . This is a generalization of the work in Ref.  in which the thermal stability of the 4D Kitaev model was analyzed by considering the dynamics of the quantum memory in contact with a thermal bath.
Let be the Hilbert space describing the system chosen as a storage media and be the algebra of operators acting on . The following definition will play an important role in this section.
Let be an observable and let be a projector onto some subspace of which is invariant under , that is, . We shall say that the observable is protected from a set of errors on a subspace iff
(Here and below we use the notation both for a subspace and the corresponding projector.) Consider as example the case when includes all single-qubit Pauli operators. Suppose for some . Then Eq. (20) implies that for all , that is, a single-qubit error cannot change the eigenvalue of for any eigenvector that belongs to . Quantum error correcting codes provide a systematic way of constructing observables protected from low-weight errors on a code-subspace, see below.
Suppose for simplicity that our goal is to encode a single qubit. We shall need a pair of observables obeying the canonical commutation rules of the Pauli operators,
In the following we shall refer to and obeying Eq. (21) as Pauli-like observables. (Note that Pauli-like observables need not to be single-qubit Pauli operators or tensor products of Pauli operators.)
Assume that the system evolves according to a Markovian master equation
where is the Lindblad operator defined by
The operators will be referred to as quantum jump operators. For any Lindblad operator , let be the set of all quantum jump operators involved in . Integrating Eq. (22) one arrives at
We shall measure the strength of using the norm
Here the maximization is over all self-adjoint operators acting on the system Hilbert space and is the trace norm of , i. e. . Note that is distinct from the spectral norm .
The following theorem is the main result of this section.
Let be an arbitrary Lindblad operator with a set of quantum jump operators such that the Gibbs state is the fixed point of , . Suppose one can choose Pauli-like observables that are protected from the set of errors on some subspace . Suppose also that , and commute with the system Hamiltonian . Then there exist TPCP encoding and decoding maps and such that
for all one-qubit states and for all .
Note that the right-hand side of Eq. (26) provides an upper bound on the precision up to which the decoded state approximates the initial state . Thus assuming that the system consists of qubits and that the norm of the Lindblad operator grows at most as we can store a qubit reliably for a time of order
We shall refer to as the storage time and to the quantity as the relaxation rate. One can envision two scenarios when the bound Eq. (28) on the storage time can be useful: (i) the relaxation rate is exponentially small as a function of , that is, for some ; (ii) the relaxation rate is only polynomially small but the degree is sufficiently large, such that grows fast with . The first scenario can be realized for systems featuring a macroscopic (growing as ) energy barrier surrounding the states orthogonal to the protected subspace . The 4D toric code model analyzed in  provides an example of such a system. The second scenario could be realized if the energy barrier grows only logarithmically as a function of as in [17, 18]. In this case the exponent is controlled by the temperature, that is, for some . If the temperature is smaller than a critical value, the relaxation rate decays sufficiently fast to yield a storage time increasing with . A polynomial increase of the storage time is also obtained in  at any temperature, from the logarithmic divergence of a self-consistent gap. It is tempting to conjecture that such system may exist in lower spatial dimensions.
The proof of Theorem 1 involves two ingredients: (i) constructing the encoding and decoding maps (see Section 6.1), and (ii) proving that the encoded states are approximate fixed points of the Lindblad operator (see Section 6.2). Our construction of encoding and decoding maps is identical to the one used by Alicki et al. in [11, 13]. It is described in Section 6.1 which can be regarded as an overview of Section IA in . The second part of the proof is presented in Section 6.2. Our approach here is quite different from the one taken in . It yields a much simpler proof and requires less assumptions about the Lindblad operator compared to  (for instance, we don’t need the detailed balance condition).
Following [11, 12, 13] we can specialize Theorem 1 to the Markovian master equation due to Davies  which describes the dynamics induced by a weak coupling between the system and a thermal bath. It involves a coupling Hamiltonian
where are some local few-qubit operators acting on the system and the operators act on the bath.
Here are the Fourier components of , that is,
One can think about as the part of transferring energy from the system to the bath. The bath temperature enters into the equation only through the function which has to obey the detailed balance condition,
The coefficient is defined as the Fourier transform of the autocorrelation function of with respect to the bath state. One can regard as a probability (per unit of time) of quantum jumps induced by the coupling operator which transfer energy from the system to the thermal bath. The detailed balance condition guarantees that the Gibbs state is a fixed point of .
It is important to discuss how the quantum jump operators depend on the original coupling operators .
For stabilizer code Hamiltonians as in Eq. (1) the time-dependent operator acts only on a few qubits since all the terms in pairwise commute and thus where includes only those terms of that act on the same qubits as . Note that has only a few Bohr frequencies since it acts only on a few qubits. It means that any quantum jump operator in the Davies master equation acts only on a few qubits and the total number of the quantum jump operators is roughly the same as the number of the coupling operators .
This issue is more subtle for subsystem codes, since may be a highly non-local operator for long times and the number of Bohr frequencies may be exponentially large. However, it is also clear that the non-locality of is only due to multiplying it with non-local elements in the gauge group . Hence remains local modulo gauge group transformations.
Let us specialize the Theorem 1 to the Davies master equation, see Eqs. (22,31). The condition that the observables and are protected from all quantum jump operators in might seem too demanding since the operators may be highly non-local, see the remark above. Fortunately, it is sufficient to require that and are protected from a set of errors including all coupling operators . Indeed, since, by assumption, commutes with and , , the condition implies for any frequency . (The same remark applies to .)
Next we need an upper bound on the norm of the Davies generator , see Eq. (31).
Assuming that for all one has
where and is the total number of terms in the interaction Hamiltonian Eq. (30).
Proof. Indeed, let be an operator such that and , see Eq. (25). Fix some and let , , and . Let us bound the trace norm of a single term
Using the bound valid for any operators we get
Here the second line used Eq. (35), convexity of the norm, and the fact that .
Let be the decomposition of into positive and negative parts, that is, and . Then
To conclude, Theorem 1 can be specialized to the Davies master equation as follows. Suppose the system interacts with a thermal bath at the inverse temperature via a Hamiltonian , where and are normalized via the condition . Suppose one can choose Pauli-like observables that are protected from any coupling operator on some subspace . Suppose that , and commute with the system Hamiltonian . Then Theorem 1 implies that a qubit can be stored in the system reliably for a time , where
Note that will be for local couplings . We will discuss how to evaluate the relaxation rate in more detail in Section 7.
6.1 Proof of Theorem 1: part I
Let us start from defining the encoding and decoding maps and . Let be the algebra generated by , , , and . For any algebra let us define the center of as
Clearly , that is, has trivial center.
Fact 1: Let be any algebra such that (i) contains the identity operator; (ii) is closed under hermitian conjugation; (iii) has a trivial center. Then there exists a (virtual) tensor product structure such that
It implies that there is a decomposition such that describes a qubit and the operators are the Pauli operators acting on , that is,
By assumption, the system’s Hamiltonian commutes with . Therefore acts trivially on and thus there exists such that
Note that . Thus the Gibbs state can be written as
Define the encoding map as
Define the decoding map formally as the partial trace over the subsystem ,
Clearly, is the identity map.
To demonstrate this formalism, let us explain how the encoding map is constructed for the special case of stabilizer (subsystem) codes. Imagine that one needs to store a single qubit state with . We encode into the thermal state where with the error-corrected logical operators . Note that commutes with .
The central idea underlying the encoding into the thermal state is that is the same as the stationary state satisfying on . Thus we can expect that if thermal fluctuations do not build up to logical errors, the state would remain close to the initial state .
Note that is quite different from the standard encoding into the the ground state subspace, for which the requirement of a Hamiltonian with finite excitation gap appears most natural. Instead, the stability criterion of Theorem 1 using the encoding in a thermal state does not explicitly involve the spectral gap. This is an interesting point, since it has become clear now that the presence of a gap does not imply robustness of topological protection. On the other hand, it might be possible to obtain a self-correcting quantum memory for a Hamiltonian with vanishing gap at large .
6.2 Proof of Theorem 1: part II
Let be any encoded state. Using Eq. (45) one check that can be represented as
Consider a family of states
Taking into account that commutes with we can represent the derivative as
Using the fact that for any TPCP map and any operator we get
Thus we have to prove an upper bound on the norm of . Inserting twice the decomposition we get
Using the identity valid for any operators , taking into account that and using Eq. (25), one easily gets
We shall bound the norm of and using the fact that . Indeed, using the assumption that and for all one can rewrite as
It follows that
Using , , and we can rewrite as