Classical-Quantum Synergies in the Theory and Practice of Quantum Error Correction

Quantum low-density parity-check (qLDPC) codes can achieve high encoding rates and good code distance scaling, providing a promising route to low-overhead fault-tolerant quantum computing. However, the long-range connectivity required to implement such codes makes their physical realization challenging. Here, we propose a hardware-efficient scheme to perform fault-tolerant quantum computation with high-rate qLDPC codes on reconfigurable atom arrays, directly compatible with recently demonstrated experimental capabilities. Our approach utilizes the product structure inherent in many qLDPC codes to implement the non-local syndrome extraction circuit via atom rearrangement, resulting in effectively constant overhead in practically relevant regimes. We prove the fault tolerance of these protocols, perform circuit-level simulations of memory and logical operations with these codes, and find that our qLDPC-based architecture starts to outperform the surface code with as few as several hundred physical qubits at a realistic physical error rate of ${10}^{-3}$. We further find that less than 3000 physical qubits are sufficient to obtain over an order of magnitude qubit savings compared to the surface code, and quantum algorithms involving thousands of logical qubits can be performed using less than ${10}^5$ physical qubits. Our work paves the way for explorations of low-overhead quantum computing with qLDPC codes at a practical scale, based on current experimental technologies.

We show how to realize a general quantum circuit involving gates between arbitrary pairs of qubits by means of geometrically local quantum operations and efficient classical computation. We prove that circuit-level local stochastic noise modeling an imperfect implementation of our derived schemes is equivalent to local stochastic noise in the original circuit. Our constructions incur a constant-factor increase in the quantum circuit depth and a polynomial overhead in the number of qubits: To execute an arbitrary quantum circuit on $n$ qubits, we give a 3D quantum fault-tolerance architecture involving $O(n^{3/2}{\log }^{3}n)$ qubits, and a quasi-2D architecture using $O(n^2{\log }^{3}n)$ qubits. Applied to recent fault-tolerance constructions, this gives a fault-tolerance threshold theorem for universal quantum computations with local operations, a polynomial qubit overhead and a quasi-polylogarithmic depth overhead. More generally, our transformation dispenses with the need for considering the locality of operations when designing schemes for fault-tolerant quantum information processing.

Quantum stabilizer codes often face the challenge of syndrome errors due to error-prone measurements. To address this issue, multiple rounds of syndrome extraction are typically employed to obtain reliable error syndromes. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between two syndrome extractions, and each syndrome measurement can be faulty. To handle these diverse error sources, we define a generalized check matrix over mixed quaternary and binary alphabets to characterize their error syndromes. This generalized check matrix leads to the creation of a Tanner graph comprising quaternary and binary variable nodes, which facilitates the development of belief propagation (BP) decoding algorithms to tackle phenomenological errors. Importantly, our BP decoders are applicable to general sparse quantum codes. Finally we extend this method to handle circuit-level noises by constructing a parity-check matrix over mixed alphabets for the syndrome extraction circuit.

Sheaf codes are linear codes with a fixed hierarchical collection of local codes, viewed as a sheaf of vector spaces on a finite topological space. Many existing codes, such as tensor product codes, Sipser-Spielman codes, and their more recent high-dimensional analogs, can be naturally represented as sheaf codes defined on simplicial and cubical complexes. We introduce a new property called maximal extendibility, which ensures that within a class of codes on the same space, we encounter as few obstructions as possible when extending local sections globally. It is possible to show that in every class of sheaf codes defined on the same space and parameterized by parity-check matrices with polynomial entries, there always exists a maximally extendable sheaf code. As it turns out, maximally extendable tensor product codes are good coboundary expanders, which allows one to generalize the recent constructions of good quantum low-density parity-check codes to more than two dimensions, and potentially could be used to attack the qLTC conjecture.

Decoding algorithms based on approximate tensor network contraction have proven tremendously successful in decoding 2D local quantum codes such as surface/toric codes and color codes, effectively achieving optimal decoding accuracy. In this work, we introduce several techniques to generalize tensor network decoding to higher dimensions so that it can be applied to 3D codes as well as 2D codes with noisy syndrome measurements (phenomenological noise or circuit-level noise). The three-dimensional case is significantly more challenging than 2D, as the involved approximate tensor contraction is dramatically less well-behaved than its 2D counterpart. Nonetheless, we numerically demonstrate that the decoding accuracy of our approach outperforms state-of-the-art decoders on the 3D surface code, both in the point and loop sectors, as well as for depolarizing noise. Our techniques could prove useful in near-term experimental demonstrations of quantum error correction, when decoding is to be performed offline and accuracy is of utmost importance. To this end, we show how tensor network decoding can be applied to circuit-level noise and demonstrate that it outperforms the matching decoder on the rotated surface code.

I discuss work, joint with Sohaib Alam, on dynamical logical qubits in the Bacon-Shor code. We choose measurement schedules on a $d\times d$ square lattice that at each round is a subset of the Bacon-Shor code checks. These measurement schedule results in a Floquet code with several dynamical logical qubits. In this talk, I briefly review Bacon-Shor subsystem codes, and then discuss the new family of Floquet codes. This work is part of a larger program trying to understand when one can define Floquet codes, when it is useful to do so, and subtleties with regard to defining their distance. The talk concludes with the statement of some specific open problems.

Twists are defects in a lattice that can be used to perform encoded computations. Three basic types of twists can be introduced in color codes: twists that permute color, charge of anyons, and domino twists that permute the charge label of an anyon with a color label. In this talk, we look at a subset of these twists from a coding theoretic viewpoint. Specifically, we present a systematic construction of charge permuting and color permuting twists in color codes. We show that by braiding alone, Clifford gates can be realized in color codes with charge permuting twists. We also present the implementation of a non-Clifford gate by state injection, thus completing the realization of a universal gate set. We finally discuss implementing single-qubit Clifford gates by a Pauli frame update and CNOT gate by braiding holes around twists in color codes with color permuting twists.

This talk discusses information rates in two quantum internet building blocks concerning quantum (conference) key distribution (QKD). We first focus on QKD based on time-entangled photon pairs. These systems extract key bits from photon arrival times and thus promise to deliver more than one bit per photon instead of polarization-entanglement QKD, where each entangled photon pair contributes at most one bit to the secret key. However, realistic photon detectors exhibit time jitter and require non-zero time to recover upon registering a photon arrival. We model and evaluate the effect of these impairments on information rates generated based on photon arrival times and ask whether time-entanglement-based QKD can live up to its promise. We next ask whether quantum network multicast can make conference key agreements more efficient. Since there is no quantum information without physical representation (e.g., by photons), the problem of quantum multicast initially seems nothing more than the multi-commodity flow problem of shipping a collection of different commodities through a shared network. However, we show that besides the apparent similarity to the multi-commodity flow problems, quantum networks, to a certain extent, behave as classical information networks. In particular, we show that lossless compression of multicast quantum states is possible and significantly reduces the link capacity requirements of the multicast.

The renormalisation decoders for Kitaev’s toric code introduced by Duclos-Cianci and Poulin exhibit one of the best trade-offs between accuracy and efficiency, with a time complexity in $n\log n$. One question that was left open is how they handle worst-case or adversarial errors, i.e., what is the order of magnitude of the smallest weight of an error pattern that will be wrongly decoded. We initiate such a study involving a simple hard-decision and deterministic version of the Duclos-Cianci and Poulin decoder.

In this breakout session, the discussion was led by Dr. Alexandru Paler and focused on the estimation of the resources needed for executing a quantum circuit that has been encoded based on surface code. More precisely, he presented a resource estimator framework that estimates the physical resources needed to execute a quantum algorithm on a modular superconducting architecture. It was shown how the requirements of a surface code-based circuit can de plotted as a space-time volume, which needs to be minimized. It is therefore key to develop a scalable optimization method for optimizing the space-volume graph. It was pointed out that this space-time volume picture does not include the magic sate distillation process. In this session, the need of improving the logical error rate by developing more accurate and faster (scalable) decoders than MWPM was also discussed.

This breakout session, led by Dr. Eleanor Rieffel, focused on the implementation of QEC codes on photonic quantum computing architectures. First, the basics of this qubit platform were introduced. In these quantum processors it is difficult to make photons to interact with each other and therefore photonic systems use measurement-based quantum computation in which high-entangled states are created. They have long-range photonic connectivity and make use of fusions for computation. In this kind of model gates are non-deterministic. It was also discussed what the error rates are for this technology. Furthermore, it was mentioned that the main issue of this technology is the interconnection with the optical fiber. Some of the papers below were discussed. We mostly focused on how to implement surface code in photonic processors, in which lattice-surgery can be used, and what the cost of encoding and performing quantum gates is.