arXiv subjects

Nicolas Delfosse

Publications and source records attributed to Nicolas Delfosse.

At least 37 records · Page 2Linked to original sources

arXiv2023

How to choose a decoder for a fault-tolerant quantum computer? The speed vs accuracy trade-off

Achieving practical quantum advantage requires a classical decoding algorithm to identify and correct faults during computation. This classical decoding algorithm must deliver both accuracy and speed, but in what combination? When is a decoder "fast enough" or "accurate enough"? In the case of surface codes, tens of decoding algorithms have been proposed, with different accuracies and speeds. However, it has been unclear how to choose the best decoder for a given quantum architecture. Should a faster decoder be used at the price of reduced accuracy? Or should a decoder sacrifice accuracy to fit within a given time constraint? If a decoder is too slow, it may be stopped upon reaching a time bound, at the price of some time-out failures and an increased failure rate. What then is the optimal stopping time of the decoder? By analyzing the speed vs. accuracy tradeoff, we propose strategies to select the optimal stopping time for a decoder for different tasks. We design a protocol to select the decoder that minimizes the spacetime cost per logical gate, for logical computation of a given depth. Our protocol enables comparison of different decoders, and the selection of an appropriate decoder for a given fault-tolerant quantum computing architecture. We illustrate our protocol for the surface code equipped with a desktop implementation of the PyMatching decoder. We estimate PyMatching is fast enough to implement thousands of logical gates with a better accuracy than physical qubits. However, we find it is not sufficiently fast to reach 10^5 logical gates, under certain assumptions, due to the decoding delay which forces qubits to idle and accumulate errors while idling. We expect further improvements to PyMatching are possible by running it on a better machine or by reducing the OS interference.

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arXiv2023

Simulation of noisy Clifford circuits without fault propagation

The design and optimization of a large-scale fault-tolerant quantum computer architecture relies extensively on numerical simulations to assess the performance of each component of the architecture. The simulation of fault-tolerant gadgets, which are typically implemented by Clifford circuits, is done by sampling circuit faults and propagating them through the circuit to check that they do not corrupt the logical data. One may have to repeat this fault propagation trillions of times to extract an accurate estimate of the performance of a fault-tolerant gadget. For some specific circuits, such as the standard syndrome extraction circuit for surface codes, we can exploit the natural graph structure of the set of faults to perform a simulation without fault propagation. We propose a simulation algorithm for all Clifford circuits that does not require fault propagation and instead exploits the mathematical structure of the spacetime code of the circuit. Our algorithm, which we name adjoint-based code (ABC) simulation, relies on the fact that propagation forward is the adjoint of propagation backward in the sense of Proposition 3 from [14]. We use this result to replace the propagation of trillions of fault-configurations by the backward propagation of a small number of Pauli operators which can be precomputed once and for all.

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arXiv2023

Splitting decoders for correcting hypergraph faults

The surface code is one of the most popular quantum error correction codes. It comes with efficient decoders, such as the Minimum Weight Perfect Matching (MWPM) decoder and the Union-Find (UF) decoder, allowing for fast quantum error correction. For a general linear code or stabilizer code, the decoding problem is NP-hard. What makes it tractable for the surface code is the special structure of faults and checks: Each X and Z fault triggers at most two checks. As a result, faults can be interpreted as edges in a graph whose vertices are the checks, and the decoding problem can be solved using standard graph algorithms such as Edmonds' minimum-weight perfect matching algorithm. For general codes, this decoding graph is replaced by a hypergraph making the decoding problem more challenging. In this work, we propose two heuristic algorithms for splitting the hyperedges of a decoding hypergraph into edges. After splitting, hypergraph faults can be decoded using any surface code decoder. Due to the complexity of the decoding problem, we do not expect this strategy to achieve a good error correction performance for a general code. However, we empirically show that this strategy leads to a good performance for some classes of LDPC codes because they are defined by low weight checks. We apply this splitting decoder to Floquet codes for which some faults trigger up to four checks and verify numerically that this decoder achieves the maximum code distance for two instances of Floquet codes.

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arXiv2023

Spacetime codes of Clifford circuits

We propose a scheme for detecting and correcting faults in any Clifford circuit. The scheme is based on the observation that the set of all possible outcome bit-strings of a Clifford circuit is a linear code, which we call the outcome code. From the outcome code we construct a corresponding stabilizer code, the spacetime code. Our construction extends the circuit-to-code construction of Bacon, Flammia, Harrow and Shi [2], revisited recently by Gottesman [16], to include intermediate and multi-qubit measurements. With this correspondence, we reduce the problem of correcting faults in a circuit to the well-studied problem of correcting errors in a stabilizer code. More precisely, a most likely error decoder for the spacetime code can be transformed into a most likely fault decoder for the circuit. We give efficient algorithms to construct the outcome and spacetime codes. We also identify conditions under which these codes are LDPC, and give an algorithm to generate low-weight checks, which can then be combined with effcient LDPC code decoders.

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arXiv2023

Efficient color code decoders in $d\geq 2$ dimensions from toric code decoders

We introduce an efficient decoder of the color code in $d\geq 2$ dimensions, the Restriction Decoder, which uses any $d$-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold $p_{\textrm{2D}} \approx 10.2\%$ on the square-octagon lattice is on a par with the toric code threshold on the square lattice.

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arXiv2022

Performance of planar Floquet codes with Majorana-based qubits

Quantum error correction is crucial for any quantum computing platform to achieve truly scalable quantum computation. The surface code and its variants have been considered the most promising quantum error correction scheme due to their high threshold, low overhead, and relatively simple structure that can naturally be implemented in many existing qubit architectures, such as superconducting qubits. The recent development of Floquet codes offers another promising approach. By going beyond the usual paradigm of stabilizer codes, Floquet codes achieve similar performance while being constructed entirely from two-qubit measurements. This makes them particularly suitable for platforms where two-qubit measurements can be implemented directly, such as measurement-only topological qubits based on Majorana zero modes (MZMs). Here, we explain how two variants of Floquet codes can be implemented on MZM-based architectures without any auxiliary qubits for syndrome measurement and with shallow syndrome extraction sequences. We then numerically demonstrate their favorable performance. In particular, we show that they improve the threshold for scalable quantum computation in MZM-based systems by an order of magnitude, and significantly reduce space and time overheads below threshold.

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arXiv2021

Almost-linear time decoding algorithm for topological codes

In order to build a large scale quantum computer, one must be able to correct errors extremely fast. We design a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and erasure. Our algorithm has a worst case complexity of $O(n α(n))$, where $n$ is the number of physical qubits and $α$ is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, $α(n) \leq 3$. We prove that our algorithm performs optimally for errors of weight up to $(d-1)/2$ and for loss of up to $d-1$ qubits, where $d$ is the minimum distance of the code. Numerically, we obtain a threshold of $9.9\%$ for the 2d-toric code with perfect syndrome measurements and $2.6\%$ with faulty measurements.

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arXiv2021

Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes

In this work we establish lower bounds on the size of Clifford circuits that measure a family of commuting Pauli operators. Our bounds depend on the interplay between a pair of graphs: the Tanner graph of the set of measured Pauli operators, and the connectivity graph which represents the qubit connections required to implement the circuit. For local-expander quantum codes, which are promising for low-overhead quantum error correction, we prove that any syndrome extraction circuit implemented with local Clifford gates in a 2D square patch of $N$ qubits has depth at least $Ω(n/\sqrt{N})$ where $n$ is the code length. Then, we propose two families of quantum circuits saturating this bound. First, we construct 2D local syndrome extraction circuits for quantum LDPC codes with bounded depth using only $O(n^2)$ ancilla qubits. Second, we design a family of 2D local syndrome extraction circuits for hypergraph product codes using $O(n)$ ancilla qubits with depth $O(\sqrt{n})$. Finally, we use circuit noise simulations to compare the performance of a family of hypergraph product codes using this last family of 2D syndrome extraction circuits with a syndrome extraction circuit implemented with fully connected qubits. While there is a threshold of about $10^{-3}$ for a fully connected implementation, we observe no threshold for the 2D local implementation despite simulating error rates of as low as $10^{-6}$. This suggests that quantum LDPC codes are impractical with 2D local quantum hardware. We believe that our proof technique is of independent interest and could find other applications. Our bounds on circuit sizes are derived from a lower bound on the amount of correlations between two subsets of qubits of the circuit and an upper bound on the amount of correlations introduced by each circuit gate, which together provide a lower bound on the circuit size.

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arXiv2021

Constant-overhead quantum error correction with thin planar connectivity

Quantum LDPC codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, naïve layouts couple many distant qubits with crossing connections which could be hard to build in hardware and could result in performance-degrading crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their Tanner graphs into a small number of planar layers. Each layer contains long-range connections which do not cross. For any CSS code with a degree-$δ$ Tanner graph, we design stabilizer measurement circuits with depth at most $(2δ+2)$ using at most $\lceil δ/2 \rceil$ layers. We observe a circuit-noise threshold of 0.28\% for a positive-rate code family using 49 physical qubits per logical qubit. For a physical error rate of $10^{-4}$, this family reaches a logical error rate of $10^{-15}$ using fourteen times fewer physical qubits than the surface code.

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arXiv2021

Improved quantum error correction using soft information

The typical model for measurement noise in quantum error correction is to randomly flip the binary measurement outcome. In experiments, measurements yield much richer information - e.g., continuous current values, discrete photon counts - which is then mapped into binary outcomes by discarding some of this information. In this work, we consider methods to incorporate all of this richer information, typically called soft information, into the decoding of quantum error correction codes, and in particular the surface code. We describe how to modify both the Minimum Weight Perfect Matching and Union-Find decoders to leverage soft information, and demonstrate these soft decoders outperform the standard (hard) decoders that can only access the binary measurement outcomes. Moreover, we observe that the soft decoder achieves a threshold 25\% higher than any hard decoder for phenomenological noise with Gaussian soft measurement outcomes. We also introduce a soft measurement error model with amplitude damping, in which measurement time leads to a trade-off between measurement resolution and additional disturbance of the qubits. Under this model we observe that the performance of the surface code is very sensitive to the choice of the measurement time - for a distance-19 surface code, a five-fold increase in measurement time can lead to a thousand-fold increase in logical error rate. Moreover, the measurement time that minimizes the physical error rate is distinct from the one that minimizes the logical performance, pointing to the benefits of jointly optimizing the physical and quantum error correction layers.

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arXiv2021

Toward a Union-Find decoder for quantum LDPC codes

Quantum LDPC codes are a promising direction for low overhead quantum computing. In this paper, we propose a generalization of the Union-Find decoder as adecoder for quantum LDPC codes. We prove that this decoder corrects all errors with weight up to An^α for some A, α > 0 for different classes of quantum LDPC codes such as toric codes and hyperbolic codes in any dimension D \geq 3 and quantum expander codes. To prove this result, we introduce a notion of covering radius which measures the spread of an error from its syndrome. We believe this notion could find application beyond the decoding problem. We also perform numerical simulations, which show that our Union-Find decoder outperforms the belief propagation decoder in the low error rate regime in the case of a quantum LDPC code with length 3600.

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arXiv2021

Union-Find Decoders For Homological Product Codes

Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a $[[4,2,2]]$ code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.

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arXiv2020

Optimization of the surface code design for Majorana-based qubits

The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits. Here, we present surface code error-correction schemes using $\textit{only}$ Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds. Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.

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arXiv2020

Short Shor-style syndrome sequences

We optimize fault-tolerant quantum error correction to reduce the number of syndrome bit measurements. Speeding up error correction will also speed up an encoded quantum computation, and should reduce its effective error rate. We give both code-specific and general methods, using a variety of techniques and in a variety of settings. We design new quantum error-correcting codes specifically for efficient error correction, e.g., allowing single-shot error correction. For codes with multiple logical qubits, we give methods for combining error correction with partial logical measurements. There are tradeoffs in choosing a code and error-correction technique. While to date most work has concentrated on optimizing the syndrome-extraction procedure, we show that there are also substantial benefits to optimizing how the measured syndromes are chosen and used. As an example, we design single-shot measurement sequences for fault-tolerant quantum error correction with the 16-qubit extended Hamming code. Our scheme uses 10 syndrome bit measurements, compared to 40 measurements with the Shor scheme. We design single-shot logical measurements as well: any logical Z measurement can be made together with fault-tolerant error correction using only 11 measurements. For comparison, using the Shor scheme a basic implementation of such a non-destructive logical measurement uses 63 measurements. We also offer ten open problems, the solutions of which could lead to substantial improvements of fault-tolerant error correction.

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arXiv2020

Optimizing Quantum Error Correction Codes with Reinforcement Learning

Quantum error correction is widely thought to be the key to fault-tolerant quantum computation. However, determining the most suited encoding for unknown error channels or specific laboratory setups is highly challenging. Here, we present a reinforcement learning framework for optimizing and fault-tolerantly adapting quantum error correction codes. We consider a reinforcement learning agent tasked with modifying a family of surface code quantum memories until a desired logical error rate is reached. Using efficient simulations with about 70 data qubits with arbitrary connectivity, we demonstrate that such a reinforcement learning agent can determine near-optimal solutions, in terms of the number of data qubits, for various error models of interest. Moreover, we show that agents trained on one setting are able to successfully transfer their experience to different settings. This ability for transfer learning showcases the inherent strengths of reinforcement learning and the applicability of our approach for optimization from off-line simulations to on-line laboratory settings.

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arXiv2020

Beyond single-shot fault-tolerant quantum error correction

Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requires $Ω(r d^2)$ parity check measurements for a distance-d code defined by r parity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using only r measurements. In this work, we demonstrate that fault-tolerant quantum error correction can be achieved using $O(d \log(d))$ measurements for any code with distance $d \geq Ω(n^α)$ for some constant $α> 0$. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer than r measurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code.

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arXiv2020

Hierarchical decoding to reduce hardware requirements for quantum computing

Extensive quantum error correction is necessary in order to scale quantum hardware to the regime of practical applications. As a result, a significant amount of decoding hardware is necessary to process the colossal amount of data required to constantly detect and correct errors occurring over the millions of physical qubits driving the computation. The implementation of a recent highly optimized version of Shor's algorithm to factor a 2,048-bits integer would require more 7 TBit/s of bandwidth for the sole purpose of quantum error correction and up to 20,000 decoding units. To reduce the decoding hardware requirements, we propose a fault-tolerant quantum computing architecture based on surface codes with a cheap hard-decision decoder, the lazy decoder, combined with a sophisticated decoding unit that takes care of complex error configurations. Our design drops the decoding hardware requirements by several orders of magnitude assuming that good enough qubits are provided. Given qubits and quantum gates with a physical error rate $p=10^{-4}$, the lazy decoder drops both the bandwidth requirements and the number of decoding units by a factor 50x. Provided very good qubits with error rate $p=10^{-5}$, we obtain a 1,500x reduction in bandwidth and decoding hardware thanks to the lazy decoder. Finally, the lazy decoder can be used as a decoder accelerator. Our simulations show a 10x speed-up of the Union-Find decoder and a 50x speed-up of the Minimum Weight Perfect Matching decoder.

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arXiv2020

A Scalable Decoder Micro-architecture for Fault-Tolerant Quantum Computing

Quantum computation promises significant computational advantages over classical computation for some problems. However, quantum hardware suffers from much higher error rates than in classical hardware. As a result, extensive quantum error correction is required to execute a useful quantum algorithm. The decoder is a key component of the error correction scheme whose role is to identify errors faster than they accumulate in the quantum computer and that must be implemented with minimum hardware resources in order to scale to the regime of practical applications. In this work, we consider surface code error correction, which is the most popular family of error correcting codes for quantum computing, and we design a decoder micro-architecture for the Union-Find decoding algorithm. We propose a three-stage fully pipelined hardware implementation of the decoder that significantly speeds up the decoder. Then, we optimize the amount of decoding hardware required to perform error correction simultaneously over all the logical qubits of the quantum computer. By sharing resources between logical qubits, we obtain a 67% reduction of the number of hardware units and the memory capacity is reduced by 70%. Moreover, we reduce the bandwidth required for the decoding process by a factor at least 30x using low-overhead compression algorithms. Finally, we provide numerical evidence that our optimized micro-architecture can be executed fast enough to correct errors in a quantum computer.

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