Minimum Weight Perfect Matching Research Articles

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.

Fundamental thresholds of realistic quantum error correction circuits from classical spin models

Mapping the decoding of quantum error correcting (QEC) codes to classical disordered statistical mechanics models allows one to determine critical error thresholds of QEC codes under phenomenological noise models. Here, we extend this mapping to admit realistic, multi-parameter noise models of faulty QEC circuits, derive the associated strongly correlated classical spin models, and illustrate this approach for a quantum repetition code with faulty stabilizer readout circuits. We use Monte-Carlo simulations to study the resulting phase diagram and benchmark our results against a minimum-weight perfect matching decoder. The presented method provides an avenue to assess fundamental thresholds of QEC circuits, independent of specific decoding strategies, and can thereby help guiding the development of near-term QEC hardware.

Pigeonhole Design: Balancing Sequential Experiments from an Online Matching Perspective

Practitioners and academics have long appreciated the benefits that experimentation brings to firms. For online web-facing firms, however, it still remains challenging in balancing covariate information when experimental units arrive sequentially in online field experiments. In this paper, we study a novel online experimental design problem, which we refer to as the "Online Blocking Problem." In this problem, experimental units with heterogeneous covariate information arrive sequentially and must be immediately assigned into either the control or the treatment group, with an objective of minimizing the total discrepancy, which is defined as the minimum weight perfect matching between the two groups. To solve this problem, we propose a novel experimental design approach, 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 units arrive at each pigeonhole, balances the number of control and treatment units 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 match-pair design and the completely randomized design. We conduct extensive simulations to study the numerical performance of the different designs and conclude with practical suggestions.

Quantum error correction with the color-Gottesman-Kitaev-Preskill code

The Gottesman-Kitaev-Preskill (GKP) code is an important type of bosonic quantum error-correcting code. Since the GKP code only protects against small shift errors in $\hat{p}$ and $\hat{q}$ quadratures, it is necessary to concatenate the GKP code with a stabilizer code for the larger error correction. In this paper, we consider the concatenation of the single-mode GKP code with the two-dimension (2D) color code (color-GKP code) on the square-octagon lattice. We use the Steane type scheme with a maximum-likelihood estimation (ME-Steane scheme) for GKP error correction and show its advantage for the concatenation. In our main work, the minimum-weight perfect matching (MWPM) algorithm is applied to decode the color-GKP code. Complemented with the continuous-variable information from the GKP code, the threshold of 2D color code is improved. If only data GKP qubits are noisy, the threshold reaches $\sigma\approx 0.59$ $(\bar{p}\approx13.3\%)$ compared with $\bar{p}=10.2\%$ of the normal 2D color code. If measurements are also noisy, we introduce the generalized Restriction Decoder on the three-dimension space-time graph for decoding. The threshold reaches $\sigma\approx 0.46$ when measurements in the GKP error correction are noiseless, and $\sigma\approx 0.24$ when all measurements are noisy. Lastly, the good performance of the generalized Restriction Decoder is also shown on the normal 2D color code giving the threshold at $3.1\%$ under the phenomenological error model.

Rectangular surface code under biased noise

To date, the surface code has become a promising candidate for quantum error correcting codes because it achieves a high threshold and is composed of only the nearest gate operations and low-weight stabilizers. Here, we have exhibited that the logical failure rate can be enhanced by manipulating the lattice size of surface codes that they can show an enormous improvement in the number of physical qubits for a noise model where dephasing errors dominate over relaxation errors. We estimated the logical error rate in terms of the lattice size and physical error rate. When the physical error rate was high, the parameter estimation method was applied, and when it was low, the most frequently occurring logical error cases were considered. By using the minimum weight perfect matching decoding algorithm, we obtained the optimal lattice size by minimizing the number of qubits to achieve the required failure rates when physical error rates and bias are provided .

Correcting Spanning Errors With a Fractal Code

The strongly correlated systems we use to realise quantum error-correcting codes may give rise to high-weight, problematic errors. Encouragingly, we can expect local quantum error-correcting codes with no string-like logical operators - such as the cubic code - to be robust to highly correlated, one-dimensional errors that span their lattice. The challenge remains to design decoding algorithms that utilise the high distance of these codes. Here, we begin the development of such algorithms by proposing an efficient decoder for the `Fibonacci code'; a two-dimensional classical code that mimics the fractal nature of the cubic code. Our iterative decoder finds a correction through repeated use of minimum-weight perfect matching by exploiting symmetries of the code. We perform numerical experiments that show our decoder is robust to one-dimensional, correlated errors. First, using a bit-flip noise model at low error rates, we find that our decoder demonstrates a logical failure rate that scales super exponentially in the linear size of the lattice. In contrast, a decoder that could not tolerate spanning errors would not achieve this rapid decay in failure rate with increasing system size. We also find a finite threshold using a spanning noise model that introduces string-like errors that stretch along full rows and columns of the lattice. These results provide direct evidence that our decoder is robust to one-dimensional, correlated errors that span the lattice.

NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs

In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in $K_{3,3}$-free graphs." In this paper, we finally settle this 30-year-old open problem. Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we obtain NC algorithms for perfect matching in any minor-closed graph family that forbids a one-crossing graph. This family includes several well-studied graph families including the $K_{3,3}$-minor-free graphs and $K_5$-minor-free graphs. Graphs in these families not only have unbounded genus, but can have genus as high as $O(n)$. Our method applies as well to several other problems related to perfect matching. In particular, we obtain NC algorithms for the following problems in any family of graphs (or networks) with a one-crossing forbidden minor: $\bullet$ Determining whether a given graph has a perfect matching and if so, finding one. $\bullet$ Finding a minimum weight perfect matching in the graph, assuming that the edge weights are polynomially bounded. $\bullet$ Finding a maximum $st$-flow in the network, with arbitrary capacities. The main new idea enabling our results is the definition and use of matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.

Decoding across the quantum low-density parity-check code landscape

We show that belief propagation combined with ordered statistics post-processing is a general decoder for quantum low density parity check codes constructed from the hypergraph product. To this end, we run numerical simulations of the decoder applied to three families of hypergraph product code: topological codes, fixed-rate random codes and a new class of codes that we call semi-topological codes. Our new code families share properties of both topological and random hypergraph product codes, with a construction that allows for a finely-controlled trade-off between code threshold and stabilizer locality. Our results indicate thresholds across all three families of hypergraph product code, and provide evidence of exponential suppression in the low error regime. For the Toric code, we observe a threshold in the range $9.9\pm0.2\%$. This result improves upon previous quantum decoders based on belief propagation, and approaches the performance of the minimum weight perfect matching algorithm. We expect semi-topological codes to have the same threshold as Toric codes, as they are identical in the bulk, and we present numerical evidence supporting this observation.

Neural Network Decoders for Large-Distance 2D Toric Codes

We still do not have perfect decoders for topological codes that can satisfy all needs of different experimental setups. Recently, a few neural network based decoders have been studied, with the motivation that they can adapt to a wide range of noise models, and can easily run on dedicated chips without a full-fledged computer. The later feature might lead to fast speed and the ability to operate at low temperatures. However, a question which has not been addressed in previous works is whether neural network decoders can handle 2D topological codes with large distances. In this work, we provide a positive answer for the toric code \cite{Kitaev2003Faulttolerantanyon}. The structure of our neural network decoder is inspired by the renormalization group decoder \cite{duclos2010fast, duclos2013fault}. With a fairly strict policy on training time, when the bit-flip error rate is lower than9%and syndrome extraction is perfect, the neural network decoder performs better when code distance increases. With a less strict policy, we find it is not hard for the neural decoder to achieve a performance close to the minimum-weight perfect matching algorithm. The numerical simulation is done up to code distanced=64. Last but not least, we describe and analyze a few failed approaches. They guide us to the final design of our neural decoder, but also serve as a caution when we gauge the versatility of stock deep neural networks. The source code of our neural decoder can be found at \cite{sourcecodegithub}.

Deep Q-learning decoder for depolarizing noise on the toric code

We present an AI-based decoding agent for quantum error correction of depolarizing noise on the toric code. The agent is trained using deep reinforcement learning (DRL), where an artificial neural network encodes the state-action Q-values of error-correcting $X$, $Y$, and $Z$ Pauli operations, occurring with probabilities $p_x$, $p_y$, and $p_z$, respectively. By learning to take advantage of the correlations between bit-flip and phase-flip errors, the decoder outperforms the minimum-weight-perfect-matching (MWPM) algorithm, achieving higher success rate and higher error threshold for depolarizing noise ($p_z = p_x = p_y$), for code distances $d\leq 9$. The decoder trained on depolarizing noise also has close to optimal performance for uncorrelated noise and provides functional but sub-optimal decoding for biased noise ($p_z \neq p_x = p_y$). We argue that the DRL-type decoder provides a promising framework for future practical error correction of topological codes, striking a balance between on-the-fly calculations, in the form of forward evaluation of a deep Q-network, and pre-training and information storage. The complete code, as well as ready-to-use decoders (pre-trained networks), can be found in the repository https://github.com/mats-granath/toric-RL-decoder.

Fault-Tolerant Thresholds for the Surface Code in Excess of 5% under Biased Noise.

Noise in quantum computing is countered with quantum error correction. Achieving optimal performance will require tailoring codes and decoding algorithms to account for features of realistic noise, such as the common situation where the noise is biased towards dephasing. Here we introduce an efficient high-threshold decoder for a noise-tailored surface code based on minimum-weight perfect matching. The decoder exploits the symmetries of its syndrome under the action of biased noise and generalizes to the fault-tolerant regime where measurements are unreliable. Using this decoder, we obtain fault-tolerant thresholds in excess of 6% for a phenomenological noise model in the limit where dephasing dominates. These gains persist even for modest noise biases: we find a threshold of ∼5% in an experimentally relevant regime where dephasing errors occur at a rate 100 times greater than bit-flip errors.

Neural ensemble decoding for topological quantum error-correcting codes

Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decoding techniques, while possibly having similar logical error rates, can perform differently on the same error syndrome. We use machine learning techniques to assign a given error syndrome to the decoder which is likely to decode it correctly. We apply our framework to an ensemble of Minimum-Weight Perfect Matching (MWPM) and Hard-Decision Re-normalization Group (HDRG) decoders for the surface code in the depolarizing noise model. Our simulations show an improvement of 38.4%, 14.6%, and 7.1% over the pseudo-threshold of MWPM in the instance of distance 5, 7, and 9 codes, respectively. Lastly, we discuss the advantages and limitations of our framework and applicability to other error-correcting codes. Our framework can provide a significant boost to error correction by combining the strengths of various decoders. In particular, it may allow for combining very fast decoders with moderate error-correcting capability to create a very fast ensemble decoder with high error-correcting capability.

Reinforcement learning for optimal error correction of toric codes

We apply deep reinforcement learning techniques to design high threshold decoders for the toric code under uncorrelated noise. By rewarding the agent only if the decoding procedure preserves the logical states of the toric code, and using deep convolutional networks for the training phase of the agent, we observe near-optimal performance for uncorrelated noise around the theoretically optimal threshold of 11%. We observe that, by and large, the agent implements a policy similar to that of minimum weight perfect matchings even though no bias towards any policy is given a priori.

Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits

In this work we introduce two code families, which we call the heavy hexagon code and heavy square code. Both code families are implemented by assigning physical data and ancilla qubits to both vertices and edges of low degree graphs. Such a layout is particularly suitable for superconducting qubit architectures to minimize frequency collisions and crosstalk. In some cases, frequency collisions can be reduced by several orders of magnitude. The heavy hexagon code is a hybrid surface/Bacon-Shor code mapped onto a (heavy) hexagonal lattice whereas the heavy square code is the surface code mapped onto a (heavy) square lattice. In both cases, the lattice includes all the ancilla qubits required for fault-tolerant error-correction. Naively, the limited qubit connectivity might be thought to limit the error-correcting capability of the code to less than its full distance. Therefore, essential to our construction is the use of flag qubits. We modify minimum weight perfect matching decoding to efficiently and scalably incorporate information from measurements of the flag qubits and correct up to the full code distance while respecting the limited connectivity. Simulations show that high threshold values for both codes can be obtained using our decoding protocol. Further, our decoding scheme can be adapted to other topological code families.

Optimal Network Topology Design in Composite Systems for Structural Controllability

This article deals with structural controllability of a linear time invariant (LTI) composite system consisting of several circuits/subsystems. We consider subsystems with structurally similar state matrices, i.e., the zero/nonzero pattern of the state matrices of the subsystems are the same, but dynamics can be different due to different numerical values. Structurally similar subsystems arise in large circuits implemented using many similar smaller circuits and agent-based networks consisting of homogeneous agents. The subsystems may not be structurally controllable individually; however, structural controllability of the composite system can be achieved by sharing state information. Sharing of information among subsystems incurs cost and our aim is to design a structurally controllable composite system with minimum information sharing. Minimizing information sharing is the same as minimizing the number of interaction links between subsystems, referred to as interconnections . An optimal network topology is one with a minimum number of interconnections. This article presents a closed-form expression for the minimum number of interconnections for structural controllability and derives a polynomial time algorithm to find an optimal network topology. The algorithm is based on a minimum weight perfect matching algorithm and a so-called edge reconstruction process. The minimum number of interconnections required is formulated in terms of two indices we define in the article: maximum commonality index ( $\alpha _{{{\mathscr N}}}$ ) and dilation index ( $\beta _{{\mathcal {I}}}$ ). Loosely speaking, more connectedness of subsystems leads to a lower total value of $\alpha _{{{\mathscr N}}}$ and $\beta _{{\mathcal {I}}}$ . Further, $\alpha _{{{\mathscr N}}}$ decreases if the subsystems have fewer numbers of connected components. We apply and verify our general result to special cases that arise in control, where the minimum number can be more directly obtained. The special cases considered are structurally cyclic subsystems, irreducible subsystems, and subsystems in controller canonical form.

Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code

Bosonic quantum error correction is a viable option for realizing error-corrected quantum information processing in continuous-variable bosonic systems. Various single-mode bosonic quantum error-correcting codes such as cat, binomial, and GKP codes have been implemented experimentally in circuit QED and trapped ion systems. Moreover, there have been many theoretical proposals to scale up such single-mode bosonic codes to realize large-scale fault-tolerant quantum computation. Here, we consider the concatenation of the single-mode GKP code with the surface code, namely, the surface-GKP code. In particular, we thoroughly investigate the performance of the surface-GKP code by assuming realistic GKP states with a finite squeezing and noisy circuit elements due to photon losses. By using a minimum-weight perfect matching decoding algorithm on a 3D space-time graph, we show that fault-tolerant quantum error correction is possible with the surface-GKP code if the squeezing of the GKP states is higher than 11.2dB in the case where the GKP states are the only noisy elements. We also show that the squeezing threshold changes to 18:6dB when both the GKP states and circuit elements are comparably noisy. At this threshold, each circuit component fails with probability 0.69%. Finally, if the GKP states are noiseless, fault-tolerant quantum error correction with the surface-GKP code is possible if each circuit element fails with probability less than 0.81%. We stress that our decoding scheme uses the additional information from GKP-stabilizer measurements and we provide a simple method to compute renormalized edge weights of the matching graphs. Furthermore, our noise model is general as it includes full circuit-level noise.

High-Threshold Code for Modular Hardware With Asymmetric Noise

We consider an approach to fault tolerant quantum computing based on a simple error detecting code operating as the substrate for a conventional surface code. We develop a customised decoder to process the information about the likely location of errors, obtained from the error detect stage, with an advanced variant of the minimum weight perfect matching algorithm. A threshold gate-level error rate of 1.42% is found for the concatenated code given highly asymmetric noise. This is superior to the standard surface code and remains so as we introduce a significant component of depolarising noise; specifically, until the latter is 70% the strength of the former. Moreover, given the asymmetric noise case, the threshold rises to 6.24% if we additionally assume that local operations have 20 times higher fidelity than long range gates. Thus for systems that are both modular and prone to asymmetric noise our code structure can be very advantageous.

Directed negative-weight percolation.

We consider the negative-weight percolation model on directed graphs. In particular, we study the model on a two-dimensional, weighted, periodic, centered square lattice. Bond weights are taken from a distribution that allows for both positive and negative values. For a given realization of the disorder, a minimally weighted directed loop and path configuration is determined by performing a nontrivial transformation of the original lattice into a minimum-weight perfect matching problem. For this problem, fast polynomial-time algorithms are available, thus we could study large systems with high accuracy. Depending on the fraction of negatively and positively weighted bonds on the lattice, a continuous phase transition is identified and its characterizing critical exponents are estimated by means of a finite-size scaling analysis. We observe a strong change of the universality class with respect to standard directed percolation as well as with respect to undirected negative-weight percolation. Furthermore, the relation to directed polymers in random media is illustrated.

Assignment problem with conflicts

We focus on an extension of the assignment problem with additional conflict (pair) constraints in conjunction with the assignment constraints and binary restrictions. Given a bipartite graph with a cost associated with each edge and a conflict set of edge pairs, the assignment problem with conflict constraints corresponds to finding a minimum weight perfect matching without any conflicting edge pair. For example, some chemicals cannot be processed on close processors, food and toxic products cannot be stored neighboring locations at the same storage area, and machines cannot be sent to process jobs without satisfying some spatial constraints. Unlike the well-known assignment problem, this problem is NP-hard. We first introduce a realistic special class and demonstrate its polynomial solvability. Then, we propose a Branch-and-Bound algorithm and a Russian Doll Search algorithm using the assignment problem relaxations for lower bound computations, and introduce combinatorial branching rules based on the conflicting edges in an optimal solution of the relaxations. According to the extensive computational experiments we can say that the proposed algorithms are very efficient.