Approximate quantum error correction (AQEC) provides a versatile framework for both quantum information processing and probing many-body entanglement. We reveal a fundamental tension between the error-correcting power of an AQEC and the hardness of code state preparation. More precisely, through a novel application of the Lovász local lemma, we establish a fundamental trade-off between local indistinguishability and circuit complexity, showing that orthogonal short-range entangled states must be distinguishable via a local operator. These results offer a powerful tool for exploring quantum circuit complexity across diverse settings. As applications, we derive stronger constraints on the complexity of AQEC codes with transversal logical gates and establish strong complexity lower bounds for W state preparation. Our framework also provides a novel perspective for systems with Lieb-Schultz-Mattis type constraints.
Approximate quantum error correction (AQEC) provides a versatile framework for both quantum information processing and probing many-body entanglement. We reveal a fundamental tension between the error-correcting power of an AQEC and the hardness of code state preparation. More precisely, through a novel application of the Lovász local lemma, we establish a fundamental trade-off between local indistinguishability and circuit complexity, showing that orthogonal short-range entangled states must be distinguishable via a local operator. These results offer a powerful tool for exploring quantum circuit complexity across diverse settings. As applications, we derive stronger constraints on the complexity of AQEC codes with transversal logical gates and establish strong complexity lower bounds for W state preparation. Our framework also provides a novel perspective for systems with Lieb-Schultz-Mattis type constraints.
Approximate quantum error correction (AQEC) extends the framework of discrete- and continuous-variable quantum error correction beyond the Knill-Laflamme (KL) conditions, where the recovery performance is quantified by entanglement fidelity. Recent studies have enabled efficient evaluation of near-optimal entanglement fidelity using transpose-channel recovery. Yet, determining the global optimal recovery map and its entanglement fidelity for general codes beyond the KL conditions remains a major computational challenge. Direct optimization becomes prohibitive as the number of noise Kraus operators grows rapidly with system size, and existing approaches lack rigorous guarantees for reducing this optimization to a tractable dimension. Here, we derive an explicit characterization of the optimal environmental state of complement channel, which transforms the optimization over recovery channels into an equivalent optimization over quotient unitaries. For a broader class of codes that satisfy only the orthogonality part of the KL conditions, we show that the optimal recovery map admits an explicit analytical form. Building on this form, we derive novel rigorous lower bounds of entanglement fidelity that strictly improve upon the transpose-recovery bound. We further develop a novel recovery strategy based on principle components, and derive a rigorous bound on the error introduced by noise truncation. Our approach enables efficient searches for approximate recovery maps for AQEC codes, avoiding the need to optimize over the full Kraus-operator space.
Quantum computers are growing in size, and design decisions are being made now that attempt to squeeze more computation out of these machines. In this spirit, we design a method to boost the computational power of nearterm quantum computers by adapting protocols used in quantum error correction to implement “Approximate Quantum Error Correction (AQEC):” By approximating fully-fledged error correction mechanisms, we can increase the compute volume (qubits × gates, or “Simple Quantum Volume (SQV)”) of near-term machines. The crux of our design is a fast hardware decoder that can approximately decode detected error syndromes rapidly. Specifically, we demonstrate a proof-of-concept that approximate error decoding can be accomplished online in near-term quantum systems by designing and implementing a novel algorithm in superconducting Single Flux Quantum (SFQ) logic technology. This avoids a critical decoding backlog, hidden in all offline decoding schemes, that leads to idle time exponential in the number of T gates in a program [58]. Our design utilizes one SFQ processing module per physical quantum bit. Employing state-of-the-art SFQ synthesis tools, we show that the circuit area, power, and latency are within the constraints of typical, contemporary quantum system designs. Under a pure dephasing error model, the proposed accelerator and AQEC solution is able to expand SQV by factors between 3,402 and 11,163 on expected near-term machines. The decoder achieves a 5% accuracy threshold as well as pseudo-thresholds of approximately 5%, 4.75%, 4.5%, and 3.5% physical error rates for code distances 3, 5, 7, and 9, respectively. Decoding solutions are achieved in a maximum of $\sim$20 nanoseconds on the largest code distances studied. By avoiding the exponential idle time in offline decoders, we achieve a 10x reduction in required code distances to achieve the same logical performance as alternative designs.
We develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill--Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters--the \emph{environment-leakage distance}, controlling worst-case performance, and the \emph{Knill-Laflamme Hellinger distance}, characterizing the average-case performance of Petz recovery.
To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric--error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.
We revisit the extendability-based semi-definite programming hierarchy introduced by Berta et al. [Mathematical Programming, 1 - 49 (2021)], which provides converging outer bounds on the optimal fidelity of approximate quantum error correction (AQEC). As our first contribution, we introduce a measurement-based rounding scheme that extracts inner sequences of certifiably good encoder-decoder pairs from this outer hierarchy. To address the computational complexity of evaluating fixed levels of the hierarchy, we investigate the use of symmetry-based dimension reduction. In particular, we combine noise symmetries - such as those present in multiple copies of the qubit depolarizing channel - with the permutational symmetry arising from the extendability of the optimization variable. This framework is illustrated through basic, but already challenging numerical examples that showcase its practical effectiveness. Our results contribute to narrowing the gap between theoretical developments in quantum information theory and their practical applications in the analysis of small-scale quantum error-correcting codes.
Entanglement renormalization can be viewed as an encoding circuit for a family of approximate quantum error correcting codes. The logical information becomes progres-sively more well-protected against erasure errors at larger length scales. In particular, an approximate variant of holographic quantum error correcting code emerges at low energy for critical systems. This implies that two operators that are largely separated in scales behave as if they are spatially separated operators, in the sense that they obey a Lieb-Robinson type locality bound under a time evolution generated by a local Hamiltonian.
The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings
(Quantum Information and Computation, 2014) -- which posits the existence of a
local Hamiltonian with a super-constant circuit lower bound on the complexity
of all low-energy states -- identifies a fundamental obstacle to the resolution
of the quantum PCP conjecture. In this paper, we provide new techniques, based
on entropic and local indistinguishability arguments, for proving circuit lower
bounds for the low-energy states of local Hamiltonians arising from quantum
error-correcting codes. For local Hamiltonians arising from nearly linear-rate and
polynomial-distance LDPC stabilizer codes, we prove a super-constant circuit
lower bounds for the complexity of all states of energy o(n) (which can be
viewed as an almost linear NLTS theorem). Such codes are known to exist and, in
general, are not locally-testable, a property previously suspected to be
essential for the NLTS conjecture. Curiously, such codes can also be
constructed on a constant-dimensional lattice, showing that complex
entanglement at high energies can persist in physically relevant systems.