Minimum Weight Perfect Matching Research Articles

Pigeonhole Design: Balancing Sequential Experiments from an Online Matching Perspective

Practitioners and academics have long appreciated the benefits of covariate balancing when they conduct randomized experiments. For web-facing firms running online A/B tests, however, it still remains challenging in balancing covariate information when experimental subjects arrive sequentially. In this paper, we study an online experimental design problem, which we refer to as the online blocking problem. In this problem, experimental subjects with heterogeneous covariate information arrive sequentially and must be immediately assigned into either the control or the treated group. The objective is to minimize the total discrepancy, which is defined as the minimum weight perfect matching between the two groups. To solve this problem, we propose a randomized design of experiment, which we refer to as the pigeonhole design. The pigeonhole design first partitions the covariate space into smaller spaces, which we refer to as pigeonholes, and then, when the experimental subjects arrive at each pigeonhole, balances the number of control and treated subjects for each pigeonhole. We analyze the theoretical performance of the pigeonhole design and show its effectiveness by comparing against two well-known benchmark designs: the matched-pair design and the completely randomized design. We identify scenarios when the pigeonhole design demonstrates more benefits over the benchmark design. To conclude, we conduct extensive simulations using Yahoo! data to show a 10.2% reduction in variance if we use the pigeonhole design to estimate the average treatment effect. This paper was accepted by George Shanthikumar, data science. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.02184 .

Efficient Syndrome Decoder for Heavy Hexagonal QECC via Machine Learning

Error syndromes for heavy hexagonal code and other topological codes such as surface code have typically been decoded by using Minimum Weight Perfect Matching– (MWPM) based methods. Recent advances have shown that topological codes can be efficiently decoded by deploying machine learning (ML) techniques, in particular with neural networks. In this work, we first propose an ML-based decoder for heavy hexagonal code and establish its efficiency in terms of the values of threshold and pseudo-threshold for various noise models. We show that the proposed ML-based decoding method achieves ~ 5 × higher values of threshold than that for MWPM. Next, exploiting the property of subsystem codes, we define gauge equivalence for heavy hexagonal code, by which two distinct errors can belong to the same error class. A linear search-based method is proposed for determining the equivalent error classes. This provides a quadratic reduction in the number of error classes to be considered for both bit flip and phase flip errors and thus a further improvement of ~ 14% in the threshold over the basic ML decoder. Last, a novel technique based on rank to determine the equivalent error classes is presented, which is empirically faster than the one based on linear search.

Ecurrent neural network decoding of rotated surface codes based on distributed strategy

Abstract Quantum error correction is a crucial technology for realizing quantum computers. These computers achieve fault-tolerant quantum computing by detecting and correcting errors using decoding algorithms. Quantum error correction using neural network-based machine learning methods is a promising approach that is adapted to physical systems without the need to build noise models. In this paper, we use a distributed decoding strategy, which effectively alleviates the problem of exponential growth of the training set required for neural networks as the code distance of quantum error-correcting codes increases. Our decoding algorithm is based on renormalization group decoding and recurrent neural network decoder. The recurrent neural network is trained through the Resnet architecture to improve its decoding accuracy. Then we tested the decoding performance of our distributed strategy decoder, recurrent neural network decoder and the classic minimum weight perfect matching(MWPM) decoder for rotated surface codes with different code distances under the circuit noise model, the thresholds of this three decoders are about 0.0052, 0.0051, and 0.0049 respectively. Our results demonstrate that the distributed strategy decoder outperforms the other two decoders, achieving approximately a 5% improvement in decoding efficiency compared to the MWPM decoder and approximately a 2% improvement compared to the recurrent neural network decoder.

Constrained many-to-many point matching in two dimensions

AbstractIn the minimum-weight many-to-many point matching problem, we are given a set R of red points and a set B of blue points in the plane, of total size N, and we want to pair up each point in R to one or more points in B and vice versa so that the sum of distances between the paired points is minimized. This problem can be solved in $$O(N^3)$$ O ( N 3 ) time by using a reduction to the minimum-weight perfect matching problem, and thus, it is not fast enough to be used for on-line systems where a large number of tunes need to be compared. Motivated by similarity problems in music theory, in this paper we study several constrained minimum-weight many-to-many point matching problems in which the allowed pairings are given by geometric restrictions, i.e., a bichromatic pair can be matched if and only if the corresponding points satisfy a specific condition of closeness. We provide algorithms to solve these constrained versions in O(N) time when the sets R and B are given ordered by abscissa.

Pipelined correlated minimum weight perfect matching of the surface code

We describe a pipeline approach to decoding the surface code using minimum weight perfect matching, including taking into account correlations between detection events. An independent no-communication parallelizable processing stage reweights the graph according to likely correlations, followed by another no-communication parallelizable stage for high confidence matching. A later general stage finishes the matching. This is a simplification of previous correlated matching techniques which required a complex interaction between general matching and re-weighting the graph. Despite this simplification, which gives correlated matching a better chance of achieving real-time processing, we find the logical error rate practically unchanged. We validate the new algorithm on the fully fault-tolerant toric, unrotated, and rotated surface codes, all with standard depolarizing noise. We expect these techniques to be applicable to a wide range of other decoders.

Quantum error correction with fractal topological codes

Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension 2+ϵ, which admits a fault-tolerant non-Clifford CCZ gate \cite{zhu2021topological}. We investigate the performance of such FSCs as fault-tolerant quantum memories. We prove that there exist decoding strategies with non-zero thresholds for bit-flip and phase-flip errors in the FSCs with Hausdorff dimension 2+ϵ. For the bit-flip errors, we adapt the sweep decoder, developed for string-like syndromes in the regular 3D surface code, to the FSCs by designing suitable modifications on the boundaries of the holes in the fractal lattice. Our adaptation of the sweep decoder for the FSCs maintains its self-correcting and single-shot nature. For the phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder for the point-like syndromes. We report a sustainable fault-tolerant threshold (∼1.7%) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by 2.95%) for the MWPM decoder for a particular FSC with Hausdorff dimension DH≈2.966. The latter can be mapped to a lower bound of the critical point of a confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.

Constrained read-once refutations in UTVPI constraint systems: A parallel perspective

Abstract In this paper, we analyze two types of refutations for Unit Two Variable Per Inequality (UTVPI) constraints. A UTVPI constraint is a linear inequality of the form: $a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{k}$ , where $a_{i},a_{j}\in \{0,1,-1\}$ and $b_{k} \in \mathbb{Z}$ . A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: ${\bf A \cdot x \le b}$ . UTVPI constraints are used in many domains including operations research and program verification. We focus on two variants of read-once refutation (ROR). An ROR is a refutation in which each constraint is used at most once. A literal-once refutation (LOR), a more restrictive form of ROR, is a refutation in which each literal ( $x_i$ or $-x_i$ ) is used at most once. First, we examine the constraint-required read-once refutation (CROR) problem and the constraint-required literal-once refutation (CLOR) problem. In both of these problems, we are given a set of constraints that must be used in the refutation. RORs and LORs are incomplete since not every system of linear constraints is guaranteed to have such a refutation. This is still true even when we restrict ourselves to UCSs. In this paper, we provide NC reductions between the CROR and CLOR problems in UCSs and the minimum weight perfect matching problem. The reductions used in this paper assume a CREW PRAM model of parallel computation. As a result, the reductions establish that, from the perspective of parallel algorithms, the CROR and CLOR problems in UCSs are equivalent to matching. In particular, if an NC algorithm exists for either of these problems, then there is an NC algorithm for matching.

A logic-based Benders decomposition solution approach for two covering problems that consider the underlying transportation

We investigate two maximal covering location problems with capacity restrictions, minimum workload, and transportation. The problems are inspired by a waste collection problem in which large waste containers are scattered throughout the municipality, and the residents bring their waste to these containers. We take the residents’ preferences into account when allocating them to locations. When a container is full, a vehicle transports an empty container from the disposal facility (depot) to that location and replaces it. We propose a mixed-integer linear programming formulation for the problems in which vehicles can carry one or two containers, and apply a logic-based Benders decomposition approach for the latter. Here, the sub problem is a multi-period minimum weight perfect matching problem. We show that our logic-based Benders decomposition approach outperforms the direct formulation in terms of solution quality and speed. We further show that transportation of two containers at a time reduces the distance to be driven by 29.5% on average, without compromising the covering level. Furthermore, we analyze the effect of imposing a minimum workload as well as the effect of changing the focus between transportation and covering.

On computing discretized Ricci curvatures of graphs: Local algorithms and (localized) fine-grained reductions

Characterizing shapes of high-dimensional objects via Ricci curvatures plays a critical role in many research areas in mathematics and physics. However, even though several discretizations of Ricci curvatures for discrete combinatorial objects such as networks have been proposed and studied by mathematicians, the computational complexity aspects of these discretizations have escaped the attention of theoretical computer scientists to a large extent. In this paper, we study one such discretization, namely the Ollivier-Ricci curvature, from the perspective of efficient computation by fine-grained reductions and local query-based algorithms. Our main contributions are the following.▹We relate our curvature computation problem to minimum weight perfect matching problem on complete bipartite graphs via fine-grained reduction.▹We formalize the computational aspects of the curvature computation problems in suitable frameworks so that they can be studied by researchers in local algorithms.▹We provide the first known lower and upper bounds on queries for query-based algorithms for the curvature computation problems in our local algorithms framework. En route, we also illustrate a localized version of our fine-grained reduction. We believe that our results bring forth an intriguing set of research questions, motivated both in theory and practice, regarding designing efficient algorithms for curvatures of geometrical objects.

Techniques for combining fast local decoders with global decoders under circuit-level noise

Implementing algorithms on a fault-tolerant quantum computer will require fast decoding throughput and latency times to prevent an exponential increase in buffer times between the applications of gates. In this work we begin by quantifying these requirements. We then introduce the construction of local neural network (NN) decoders using three-dimensional convolutions. These local decoders are adapted to circuit-level noise and can be applied to surface code volumes of arbitrary size. Their application removes errors arising from a certain number of faults, which serves to substantially reduce the syndrome density. Remaining errors can then be corrected by a global decoder, such as Blossom or union find, with their implementation significantly accelerated due to the reduced syndrome density. However, in the circuit-level setting, the corrections applied by the local decoder introduce many vertical pairs of highlighted vertices. To obtain a low syndrome density in the presence of vertical pairs, we consider a strategy of performing a syndrome collapse which removes many vertical pairs and reduces the size of the decoding graph used by the global decoder. We also consider a strategy of performing a vertical cleanup, which consists of removing all local vertical pairs prior to implementing the global decoder. By applying our local NN decoder and the vertical cleanup strategy to a d = 17 surface code volume, we show a speedup of the minimum-weight perfect matching decoder. Lastly, we estimate the cost of implementing our local decoders on field programmable gate arrays.

Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code

Bosonic codes provide an alternative option for quantum error correction. An important category of bosonic codes called the Gottesman-Kitaev-Preskill (GKP) code has aroused much interest recently. Theoretically, the error correction ability of GKP code is limited since it can only correct small shift errors in position and momentum quadratures. A natural approach to promote the GKP error correction for large-scale, fault-tolerant quantum computation is concatenating encoded GKP states with a stabilizer code. The performance of the XZZX surface-GKP code, i.e., the single-mode GKP code concatenated with the XZZX surface code is investigated in this paper under two different noise models. Firstly, in the code-capacity noise model, the asymmetric rectangular GKP code with parameter $\lambda$ is introduced. Using the minimum weight perfect matching decoder combined with the continuous-variable GKP information, the optimal threshold of the XZZX-surface GKP code reaches $\sigma\approx0.67$ when $\lambda=2.1$, compared with the threshold $\sigma\approx0.60$ of the standard surface-GKP code. Secondly, we analyze the shift errors of two-qubit gates in the actual implementation and build the full circuit-level noise model. By setting the appropriate bias parameters, the logical error rate is reduced by several times in some cases. These results indicate the XZZX surface-GKP codes are more suitable for asymmetric concatenation under the general noise models. We also estimate the overhead of the XZZX-surface GKP code which uses about 291 GKP states with the noise parameter 18.5 dB ($\kappa/g \approx 0.71\%$) to encode a logical qubit with the error rate $2.53\times10^{-7}$, compared with the qubit-based surface code using 3041 qubits to achieve almost the same logical error rate.

Approximate error correction scheme for three-dimensional surface codes based reinforcement learning

Quantum error correction technology is an important method to eliminate errors during the operation of quantum computers. In order to solve the problem of influence of errors on physical qubits, we propose an approximate error correction scheme that performs dimension mapping operations on surface codes. This error correction scheme utilizes the topological properties of error correction codes to map the surface code dimension to three dimensions. Compared to previous error correction schemes, the present three-dimensional surface code exhibits good scalability due to its higher redundancy and more efficient error correction capabilities. By reducing the number of ancilla qubits required for error correction, this approach achieves savings in measurement space and reduces resource consumption costs. In order to improve the decoding efficiency and solve the problem of the correlation between the surface code stabilizer and the 3D space after dimension mapping, we employ a reinforcement learning (RL) decoder based on deep Q-learning, which enables faster identification of the optimal syndrome and achieves better thresholds through conditional optimization. Compared to the minimum weight perfect matching decoding, the threshold of the RL trained model reaches 0.78%, which is 56% higher and enables large-scale fault-tolerant quantum computation.

Fault-tolerant quaternary belief propagation decoding based on a neural network

The article discusses the challenge of finding an efficient decoder for quantum error correction codes for fault-tolerant experiments in quantum computing. The study aims to develop a better decoding scheme based on the flag-bridge fault tolerance experiment. The research compares two decoding algorithms, a deep neural network decoding scheme and a simple decoder, and a recurrent neural network decoding scheme based on the belief propagation algorithm variant MBP4 algorithm. The study improved the syndrome extraction circuit based on the flag-bridge method to meet the requirements of fault-tolerant experiments better. Two decoding schemes were studied, a combination of a deep neural network and a simple decoder and a recurrent neural network structure based on the MBP4 algorithm. The first scheme used neural networks to assist simple decoders in determining whether additional logical corrections need to be added. The second scheme used a recurrent neural network structure designed through the variant MBP4 algorithm, along with a post-processing method to pinpoint the error qubit position for decoding. Experimental results showed that the decoding scheme developed in the study improved the pseudo-threshold by 39.52% compared to the minimum-weight perfect matching decoder. The two decoders had thresholds of approximately 15.8% and 16.4%, respectively. The study’s findings suggest that the proposed decoding schemes could improve quantum error correction and fault-tolerant experiments in quantum computing.

Performance of surface codes in realistic quantum hardware

Surface codes are generally studied based on the assumption that each of the qubits that make up the surface code lattice suffers noise that is independent and identically distributed (i.i.d.). However, real benchmarks of the individual relaxation ($T_1$) and dephasing ($T_2$) times of the constituent qubits of state-of-the-art quantum processors have recently shown that the decoherence effects suffered by each particular qubit actually vary in intensity. In consequence, in this article we introduce the independent non-identically distributed (i.ni.d.) noise model, a decoherence model that accounts for the non-uniform behaviour of the docoherence parameters of qubits. Additionally, we use the i.ni.d model to study how it affects the performance of a specific family of Quantum Error Correction (QEC) codes known as planar codes. For this purpose we employ data from four state-of-the-art superconducting processors: ibmq\_brooklyn, ibm\_washington, Zuchongzhi and Rigetti Aspen-M-1. Our results show that the i.i.d. noise assumption overestimates the performance of surface codes, which can suffer up to $95\%$ performance decrements in terms of the code pseudo-threshold when they are subjected to the i.ni.d. noise model. Furthermore, we consider and describe two methods which enhance the performance of planar codes under i.ni.d. noise. The first method involves a so-called re-weighting process of the conventional minimum weight perfect matching (MWPM) decoder, while the second one exploits the relationship that exists between code performance and qubit arrangement in the surface code lattice. The optimum qubit configuration derived through the combination of the previous two methods can yield planar code pseudo-threshold values that are up to $650\%$ higher than for the traditional MWPM decoder under i.ni.d. noise.

Multidimensional Bose quantum error correction based on neural network decoder

Boson quantum error correction is an important means to realize quantum error correction information processing. In this paper, we consider the connection of a single-mode Gottesman-Kitaev-Preskill (GKP) code with a two-dimensional (2D) surface (surface-GKP code) on a triangular quadrilateral lattice. On the one hand, we use a Steane-type scheme with maximum likelihood estimation for surface-GKP code error correction. On the other hand, the minimum-weight perfect matching (MWPM) algorithm is used to decode surface-GKP codes. In the case where only the data GKP qubits are noisy, the threshold reaches σ ≈ 0.5 (bar{p}approx 12.3 %). If the measurement is also noisy, the threshold is reached σ ≈ 0.25 (bar{p}approx 10.02 %). More importantly, we introduce a neural network decoder. When the measurements in GKP error correction are noise-free, the threshold reaches σ ≈ 0.78 (bar{p}approx 15.12 %). The threshold reaches σ ≈ 0.34 (bar{p}approx 11.37 %) when all measurements are noisy. Through the above optimization method, multi-party quantum error correction will achieve a better guarantee effect in fault-tolerant quantum computing.

Practical and scalable decoder for topological quantum error correction with an Ising machine

Quantum error correction is one of the most important milestones for the realization of large-scale quantum computation. To achieve this, it is essential not only to integrate a large number of qubits with high fidelity but also to build a scalable classical system that can perform error correction. Here, we propose an efficient and scalable decoder for quantum error correction using ``Fujitsu Digital Annealer'' (DA). Specifically, the error correction problem of stabilizer codes is mapped into an Ising-type optimization problem, so-called quadratic unconstrained binary optimization (QUBO) problem, which is solved by DA. In particular, we implement the proposed DA decoder for the surface code and perform detailed numerical experiments for various code distances to see its performance and scalability. We observe that computational scaling for the DA decoder has a lower order of polynomials than the decoding methods using simulated annealing (SA) and minimum-weight perfect matching (MWPM) algorithm under all tested conditions. It is also shown that the DA decoder has advantages over the Union-Find (UF) decoder from a variety of perspectives including hardware implementation. Furthermore, the threshold behavior of the logical error probability for the DA decoder is analyzed and the resultant threshold lies between 9.4% and 9.8%, which is very close to that obtained by the MWPM decoder. This result clearly shows the high potential of the DA decoder for quantum error correction.

Multiparticle quantum walk–based error correction algorithm with two-lattice Bose–Hubbard model

When the evolution of discrete time quantum walk is carried out for particles, the ramble state is prone to error due to the influence of system noise. A multiparticle quantum walk error correction algorithm based on the two-lattice Bose–Hubbard model is proposed in this study. First, two point Bose–Hubbard models are constructed according to the local Euclidean generator, and it is proved that the two elements in the model can be replaced arbitrarily. Second, the relationship between the transition intensity and entanglement degree of the particles in the model is obtained by using the Bethe hypothesis method. Third, the position of the quantum lattice is coded and the quantum state exchange gate is constructed. Finally, the state replacement of quantum walk on the lattice point is carried out by switching the walker to the lattice point of quantum error correction code, and the replacement is carried out again. The entanglement of quantum particles in the double-lattice Bose–Hubbard model is simulated numerically. When the ratio of the interaction between particles and the transition intensity of particles is close to 0, the entanglement operation of quantum particles in the model can be realized by using this algorithm. According to the properties of the Bose–Hubbard model, quantum walking error correction can be realized after particle entanglement. This study introduces the popular restnet network as a training model, which increases the decoding speed of the error correction circuit by about 33%. More importantly, the lower threshold limit of the convolutional neural network (CNN) decoder is increased from 0.0058 under the traditional minimum weight perfect matching (MWPM) to 0.0085, which realizes the stable progress of quantum walk with high fault tolerance rate.

PyMatching: A Python Package for Decoding Quantum Codes with Minimum-Weight Perfect Matching

This article introduces PyMatching, a fast open-source Python package for decoding quantum error-correcting codes with the minimum-weight perfect matching (MWPM) algorithm. PyMatching includes the standard MWPM decoder as well as a variant, which we call local matching , that restricts each syndrome defect to be matched to another defect within a local neighborhood. The decoding performance of local matching is almost identical to that of the standard MWPM decoder in practice, while reducing the computational complexity. We benchmark the performance of PyMatching, showing that local matching is several orders of magnitude faster than implementations of the full MWPM algorithm using NetworkX or Blossom V for problem sizes typically considered in error correction simulations. PyMatching and its dependencies are open-source, and it can be used to decode any quantum code for which syndrome defects come in pairs using a simple Python interface. PyMatching supports the use of weighted edges, hook errors, boundaries and measurement errors, enabling fast decoding, and simulation of fault-tolerant quantum computing.

A real polynomial for bipartite graph minimum weight perfect matchings

In a recent paper, Beniamini and Nisan [4] gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph G⊆Kn,n has a perfect matching, together with an efficient algorithm for computing the coefficients of the monomials of this polynomial. We give the following generalization: Given an arbitrary weight function w on the edges of Kn,n, consider its set of minimum weight perfect matchings. We give the real multilinear polynomial for the Boolean function which determines if a graph G⊆Kn,n contains one of these minimum weight perfect matchings. Finally, we discuss a number of open problems which follow from [4] and our work; in particular, extending the main theorem of [4] to non-bipartite graphs.

Scalable Neural Decoder for Topological Surface Codes.

With the advent of noisy intermediate-scale quantum (NISQ) devices, practical quantum computing has seemingly come into reach. However, to go beyond proof-of-principle calculations, the current processing architectures will need to scale up to larger quantum circuits which will require fast and scalable algorithms for quantum error correction. Here, we present a neural network based decoder that, for a family of stabilizer codes subject to depolarizing noise and syndrome measurement errors, is scalable to tens of thousands of qubits (in contrast to other recent machine learning inspired decoders) and exhibits faster decoding times than the state-of-the-art union find decoder for a wide range of error rates (down to 1%). The key innovation is to autodecode error syndromes on small scales by shifting a preprocessing window over the underlying code, akin to a convolutional neural network in pattern recognition approaches. We show that such a preprocessing step allows to effectively reduce the error rate by up to 2 orders of magnitude in practical applications and, by detecting correlation effects, shifts the actual error threshold up to fifteen percent higher than the threshold of conventional error correction algorithms such as union find or minimum weight perfect matching, even in the presence of measurement errors. An insitu implementation of such a machine learning-assisted quantum error correction will be a decisive step to push the entanglement frontier beyond the NISQ horizon.