General conditions for approximate quantum error correction and near-optimal recovery channels
We derive necessary and sufficient conditions for the approximate correctability of a quantum code, generalizing the Knill-Laflamme conditions for exact error correction. Our measure of success of the recovery operation is the worst-case entanglement fidelity of the overall process. We show that the optimal recovery fidelity can be predicted exactly from a dual optimization problem on the environment causing the noise. We use this result to obtain an easy-to-calculate estimate of the optimal recovery fidelity as well as a way of constructing a class of near-optimal recovery channels that work within twice the minimal error. In addition to standard subspace codes, our results hold for subsystem codes and hybrid quantum-classical codes.
Introduction — Given the extreme fragility of quantum coherence, quantum error-correction procedures are believed to be essential for the successful implementation of quantum communication or computation. Exact correctability is characterized in general terms by the Knill-Laflamme (KL) conditions knill97 which specify the set of correctable errors for a particular code. However, practically useful codes need not be exactly correctable for any given noise model. In fact, a few exceptional examples show that allowing for a negligible error in the recovery can lead to surprisingly better codes leung97; crepeau05. This indicates that assuming exact correctability is too strong a restriction. It is therefore of considerable interest to find appropriately weaker error-correction conditions.
In this letter, we generalize the KL conditions to the case of approximate correctability. We view the KL conditions as a statement about the information gathered by the environment causing the noise beny09x1 (a fact previously noted in Ref. ogawa05). Thus our analysis makes essential use of the concept of complementary channel devetak05. Together with tools introduced in Ref. kretschmann08, this provides the basis for our main technical result (Theorem 1), which we use to obtain easily computable estimates of the optimal recovery error. We also propose a class of near-optimal recovery channels, which offers a significant simplification to the problem of finding an optimal recovery operation yamamoto05; reimpell05; fletcher08; kosut08.
The analysis of approximate error correction depends on the figure of merit used to compare the states after correction to the input states. In this work we focus on the entanglement fidelity minimized over all input states (also known as worst-case entanglement fidelity). The entanglement fidelity schumacher96 has been shown to be the pertinent fidelity measure in both quantum communication and computation scenarios since it estimates not only how well the state of the system under correction is preserved but also how its entanglement with auxiliary systems is maintained. Minimization over all inputs is essential if one is interested in guaranteeing a given fidelity when the state to be corrected is not known, as in the case of quantum computing. In contrast, most previous work has considered input-dependent fidelities barnum00; schumacher01; tyson09x1; buscemi08. Sufficient conditions for approximate correctability under the worst-case entanglement fidelity were proposed in Ref. doddamane09. Here we obtain both sufficient and necessary conditions which are a direct generalization of the KL conditions. Moreover, we prove our result in a very general context; namely for the approximation of any channel, not necessarily the identity map on the code. One advantage of this generality is that our results apply directly to the more general schemes of subsystem, or operator quantum error correction kribs05; kribs06; poulin05x1; beny07x1. The present results are also strictly stronger than those of Ref. kretschmann08x1; beny09x1 which are based on the diamond-norm distance rather than the fidelity.
Background — The problem of quantum error correction can be formulated as follows: we are given a channel which can represent either a communication channel or the open dynamics of a physical system which we would like to use as a quantum memory. The goal is to find an encoding operation and a decoding (or recovery) operation , such that the full operation is equal to the identity map. One usually assumes that the encoding is of the form where is an isometry embedding a small Hilbert space (the code) into the larger physical Hilbert space on which acts.
Given and , the KL conditions knill97 provide a simple way of testing whether a recovery channel exists. In addition, these conditions help reasoning about error correction. For instance, one can use them together with the no-cloning theorem to easily demonstrate that it is not possible to encode a qubit in qubits and faithfully decode it if or more arbitrary qubit errors occur. The reason we mention this particular example is that it is known to fail dramatically if we allow for an arbitrarily small reconstruction error (provided is large enough). Indeed, it was shown in Ref. crepeau05 that one can encode quantum information in qubits undergoing almost arbitrary errors and correct it with vanishing error as .
Here we study what becomes of the KL conditions when we allow for imperfect reconstruction of the code. Additionally, partly because it reveals an important symmetry of the problem, we also generalize quantum error correction in a different direction. We seek a “recovery” operation such that is close not necessarily to the identity on the code, but to a fixed arbitrary channel . In particular, this means that our theory applies to subsystem codes kribs05; kribs06, and more generally algebraic codes beny07x1 (representing hybrid quantum-classical information), when projects on an algebra beny09x1. Note that since we will never separate from the encoding , we will simply work with a channel “” which one can think of as . It typically maps states on a small (logical) Hilbert space to states on a larger (physical) one.
We will make essential use of the fact that a general quantum operation, or channel , can always be viewed as resulting from a unitary interaction with an “environment” whose initial state is known and which is later discarded (traced out). It does not matter which state we use since the difference can be absorbed in the unitary. What matters is the isometry defined by so that This isometry is not unique, but unique up to a further local unitary map on the environment, eventually followed by an embedding into a larger environment. From the isometry , one obtains the channel elements of simply by writing the partial trace explicitly in terms of a basis of . If instead of tracing out the environment after the unitary interaction, we trace out the target system , we obtain a channel which is said to be complementary to : . It is easy to see that
All complementary channels correspond to some choice of the orthonormal family of states in the environment.
Main result — Let be the fidelity uhlmann76 between states and . For a given state , we introduce the “entanglement fidelity” between channels and ,
where is a purification of . When , this quantity reduces to Schumacher’s entanglement fidelity of schumacher96. We will compare channels using the worst-case entanglement fidelity
which was studied in Ref. gilchrist05. We remark that relates to in the same way that the diamond-norm distance Kitaev97 relates to the trace distance. Its operational meaning can be deduced from that of Dodd01; Fuchs96.
If and are channels complementary to and , respectively, then
where the maxima are over all quantum channels with the appropriate source and target spaces.
The proof closely follows arguments used in kretschmann08. Let be the isometry for which and , and be the isometry yielding both and in the same way. Note that for a fixed state , any channel can be written as for some unitary and appropriate “environment” . Using this fact and applying Uhlmann’s theorem uhlmann76 which allows us to write the entanglement fidelity in terms of an overlap maximized over unitary operators , we obtain
where can be expressed in terms of a circuit:
where the left half circles represent input states, while the right half circles are states which are scalar multiplied with the corresponding outputs. Hence the picture represents a complex number. The wires labeled and represent the target systems for and respectively, and and are the respective “environments”. The state in the picture is arbitrary, and can be any purification of . If we reflect the picture with respect to a vertical axis through the middle, Hermitian conjugating each operator [this amounts to a complex conjugation of ], and exchange the wire labels and , and and , we see that we also have , where now is the unitary defining while comes from Uhlmann’s expression for the fidelity. Hence we just have to show that we can exchange the maximizations over and in Eq. (4). First, using the strong concavity of the fidelity, it can be shown that the over in Eq. (4) can as well be taken over the convex set of operators with operator norm . Next, note that for some operator . We know that . Since the optimal value of is real, we only need to optimize which is linear in and over the real numbers. In addition, the max over can also be taken over operators in the unit ball since then . We can now apply Shiffman’s minimax theorem grossinho01 which says that we can exchange the rightmost min and max provided that the function is convex-concave in the two arguments (in this case it is bilinear), and that the variables are optimized over convex sets. Hence, we obtain , where . ∎
Note that Eq. (3) can be seen as a necessary and sufficient condition for approximate correctability: for a given , there exists a channel such that , iff there exists a channel such that . We will see that for a large class of problems of interest the existence of is much easier to establish than that of .
Knill-Laflamme conditions — Consider the case . An example of a channel complementary to the identity is the trace: , whose target is one-dimensional. A channel whose source is one-dimensional outputs a single state. Hence, , , where is a fixed state. Theorem 1 thus says that iff , . Explicitly, suppose that the channel consists of an encoding specified by an isometry followed by noise with channel elements . In terms of matrix components, the condition that be a constant channel with output reads , where . One obtains the most familiar form of the KL conditions by expressing these equations using the projector on the code:
More generally, if projects on an algebra , then we obtain the general correctability conditions for an algebra beny07x1, namely for all . This can be shown by noting that , where is the commutant of , i.e. the set of operators commuting with all (see beny09x1 for more details.) In particular, when the algebra consists of all operators acting on a subsystem, this yields the conditions for the correctability of a subsystem code kribs06.
Let us show explicitly how Theorem 1 can be understood as a perturbation of the KL conditions in the case . Since later in our analysis we will use triangle inequalities, it is convenient to measure the error of imperfect recovery by a fidelity-based distance function. We will consider the Bures distance bures69 based on the entanglement fidelity, Note that
satisfies the triangle inequality:
We will say that a code characterized by the encoding map is -correctable under the noise channel , if there exists a recovery channel such that .
A code defined by the projector is -correctable under a noise channel , if and only if
where are the components of a density operator, and where and .
It is not a priori clear how useful this condition can be since it does not specify how to find an optimal set of coefficients . We will now show, in a more general setting, that we can find a whole set of explicit guesses for which are guaranteed to yield a value of which is less than twice the optimal one. Explicitly, this is the case whenever for some state .
Near-optimal correction — We saw that in the exact case (fidelity one), Theorem 1 yields the necessary and sufficient conditions for all quantum error-correction schemes when projects on an algebra. Here we want to show that it also yields useful conditions for the approximate version of these schemes. The problem is that in general it may not be easier to compute than . However, we will show that when is a projection, i.e. it satisfies (which is the case for error correction), we can guess a whole class of channels for which yields a good approximation to the optimal worst-case fidelity . Moreover, we can build the corresponding near-optimal recovery channels .
Suppose that . Then