Quantum capacity of the qubit depolarizing channel and the optimal asymptotic rate of quantum error-correcting codes

Calderbank and Shor's 1996 paper 'Good quantum error-correcting codes exist' proved, by a non-constructive Gilbert–Varshamov-type counting argument over weakly self-dual classical codes, that quantum error-correcting codes correcting $t$ arbitrary single-qubit errors on $n$ qubits exist with asymptotic rate $1-2H_2(2t/n)$, and compared this constructive rate with two upper bounds obtained from the Levitin–Holevo theorem and from entanglement-transmission bounds, remarking that the bounds do not meet and that the Levitin–Holevo bound is likely not tight for quantum information. Three decades later the central quantitative questions are still open: the exact quantum capacity of the qubit depolarizing channel (the capacity threshold is currently bracketed only by $\approx 0.194 \le p_{\mathrm{th}} \le 1/4$), the asymptotic optimality of the quantum Gilbert–Varshamov rate for binary stabilizer codes, and the existence of explicit, efficiently constructible quantum code families attaining these bounds. The fault-tolerance and structured-correlated-noise issues raised alongside them have been substantially resolved by the quantum accuracy threshold theorem and its extensions, so the surviving core is the capacity/tightness and constructive-attainment problem.

Background

A qubit is a two-level quantum system; a register of $n$ qubits spans a $2^n$-dimensional complex state space. Decoherence — unwanted interaction with an environment — corrupts quantum information in a way that cannot be repaired by copying, because the no-cloning theorem forbids duplicating an unknown quantum state. A quantum error-correcting code therefore encodes $k$ logical qubits into $n$ physical qubits so that the original state can be recovered after noise acts on some of the physical qubits. Shor's nine-qubit code (1995) was the first example; Calderbank and Shor (1996) and, independently, Steane (1996) then showed how to build quantum codes from classical binary linear codes, a construction now called CSS (Calderbank–Shor–Steane) codes. A binary linear code is a subspace of $\mathbb{F}_2^n$ (the space of length-$n$ bit vectors); its dimension $k$ and length $n$ give the rate $k/n$, and its minimum Hamming distance $d$ — the minimum number of coordinates in which two codewords differ — determines how many errors it corrects. Calderbank and Shor used weakly self-dual codes, meaning a code contained in its dual (the set of vectors orthogonal to all codewords), from which the two error types of the quantum code are jointly described. The Gilbert–Varshamov (GV) bound is the classical statement that codes of rate $1-H_2(d/n)$ with minimum distance $d$ exist, where $H_2(p)=-p\log_2 p-(1-p)\log_2(1-p)$ is the binary entropy; its standard proof is a greedy counting argument showing existence without producing a code. By adapting the counting to weakly self-dual codes, Calderbank and Shor proved that quantum codes correcting $t$ arbitrary errors on $n$ qubits — arbitrary decoherence of any $t$ qubits, not necessarily independent — exist with asymptotic rate $1-2H_2(2t/n)$, and noted that the argument is non-constructive.

The stochastic benchmark is the qubit depolarizing channel, which transmits a qubit unchanged with probability $1-p$ and applies each of the Pauli operators $X$ (bit flip), $Z$ (phase flip), or $Y=XZ$ with probability $p/3$; equivalently it replaces the qubit by the maximally mixed state with probability $4p/3$. The quantum capacity $Q$ of a channel is the supremum rate at which unknown quantum states — equivalently, entanglement — can be transmitted with error tending to zero over asymptotically many uses. Devetak (2005) later proved that $Q$ equals the regularized coherent information (a maximum, over arbitrarily many channel uses, of the entropy difference between channel output and environment), and that random stabilizer codes — a Pauli-group-based generalization of CSS codes — achieve the hashing bound $Q\ge 1-H_2(p)-p\log_2 3$, which is positive for $p\lesssim 0.1893$. Calderbank and Shor, writing before this theory, compared their constructive rate for the depolarizing channel with two upper bounds: one derived from the Levitin–Holevo bound (the Holevo bound on the classical information carried by quantum signals) and one from bounds on transmittable entanglement; the bounds did not coincide with their rate, and they wrote that the Levitin–Holevo bound 'is achievable for classical information, but we believe it is unlikely to be tight for quantum information'. Two further facts frame the modern question. First, quantum codes can be degenerate — distinct correctable errors can act identically on the code space, so more errors are correctable than syndrome counting suggests — and degenerate codes are known to beat the hashing bound on the depolarizing channel. Second, by the no-cloning theorem the depolarizing channel has zero quantum capacity for $p\ge 1/4$. The paper also expressed the hope that its techniques would eventually reduce the error rates required for large-scale quantum computation; that hope was later made precise and resolved by the quantum accuracy threshold theorem (Aharonov–Ben-Or; Knill–Laflamme–Zurek), which proves fault-tolerant quantum computation is possible below a constant per-component error threshold.

Problem Statement

Close the quantitative gap that Calderbank and Shor (1996) left open between their constructive rate and their capacity upper bounds for quantum communication through depolarizing-type noise, in three linked facets of one objective — establishing the true asymptotic limit of quantum error correction and whether that limit is constructively attainable.

(1) Capacity and converse. Determine the quantum capacity $Q(D_p)$ of the qubit depolarizing channel $D_p(\rho)=(1-p)\rho+\tfrac{p}{3}\left(X\rho X+Y\rho Y+Z\rho Z\right)$ for every noise level $p$, or at minimum the exact capacity threshold $p_{\mathrm{th}}=\sup\{p:Q(D_p)>0\}$, currently bracketed only by $\approx 0.194 \le p_{\mathrm{th}} \le 1/4$. This settles whether the Levitin–Holevo-type upper bound invoked in the 1996 paper is or is not tight for quantum information (the authors conjectured it is not), whether the entanglement-transmission upper bound can be improved, and whether a tight converse for the paper's constructive lower bound exists.

(2) Optimality of the quantum Gilbert–Varshamov tradeoff. Determine whether the rate $1-2H_2(2t/n)$ for quantum codes correcting $t$ arbitrary errors on $n$ qubits is asymptotically optimal for binary stabilizer/CSS codes — no matching converse is currently known — or whether degenerate codes can asymptotically exceed it, as they already exceed the corresponding hashing bound for the channel.

(3) Explicit constructions. Give an explicit, efficiently constructible family of binary quantum codes, with efficient encoding and decoding, that attains these asymptotic bounds — the rate–error tradeoff that facets (1) and (2) identify as the limit; for the adversarial model this is the quantum Gilbert–Varshamov rate $1-2H_2(2t/n)$, a function of the error fraction $t/n$; for the depolarizing channel it is the separate hashing-bound rate $1-H_2(p)-p\log_2 3$, a function of the error probability $p$. The 1996 existence proof is a non-constructive greedy counting argument over random weakly self-dual classical codes, and even classically no explicit binary code family attaining the Gilbert–Varshamov bound is known, so making the 1996 existence result effective is an open part of the same objective.

Facets (1) and (2) are the channel (stochastic) and code (adversarial) forms of a Gilbert–Varshamov-type rate–error tradeoff, given by the two distinct bounds above; facet (3) asks that the limit identified in each model be achieved by explicit constructions rather than by a non-constructive counting argument.

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