Stabilizer Codes Research Articles

Hexagonal matching codes with two-body measurements

Matching codes are stabilizer codes based on Kitaev’s honeycomb lattice model. The hexagonal form of these codes are particularly well-suited to the heavy-hexagon device layouts currently pursued in the hardware of IBM quantum. Here we show how the stabilizers of the code can be measured solely through two-body measurements that are native to the architecture. Though the subsystem code formed by these measurements has a trivial code space, the sequence in which they are measured allows the desired logical subspace to be preserved. This therefore achieves a result similar to the recently introduced Floquet codes, but via a completely different method. The process is then run on 27 and 65 qubit devices, to compare results with simulations for a standard error model. It is found that the results correspond well to simulations where the noise strength is similar to that found in the benchmarking of the devices. The best devices show results consistent with a noise model with an error probability of around 1.5%–2%.

A Combinatorial Interpretation for the Shor-Laflamme Weight Enumerators of CWS Codes

We show that one of the Shor-Laflamme weight enumerators of a codeword stabilized quantum code may be interpreted as the distance enumerator of an associated classical code.

Modeling of saturated external MHD instabilities in tokamaks: A comparison of 3D free boundary equilibria and nonlinear stability calculations

3D free boundary equilibrium computations have recently been used to model external kinks and edge harmonic oscillations (EHOs), comparing with linear MHD stability codes, and nonlinear analytic theory [Kleiner et al., Phys. Plasma Controlled Fusion 61, 084005 (2019)]. In this study, results of the VMEC equilibrium code are compared further with nonlinear reduced MHD simulations, using the JOREK code. The purpose of this investigation was to understand the extent to which the modeling approaches agree, and identify the important physical effects, which can modify the dynamics. For the simulated external kink, which is dominated by a single toroidal harmonic, good agreement is found when a large Lundquist number is used in the JOREK simulation, such that resistive effects are sub-dominant. Modeling EHOs where multiple toroidal harmonics are linearly unstable, the saturated perturbation observed can differ in the dominant toroidal harmonic. On the ideal timescale, a n = 2 EHO is observed in JOREK, while the saturated perturbation predicted by VMEC is a n = 1 mode. Extending simulations into timescales where resistive effects can play a role, similar n = 1 perturbations can be found. The coupling of different linearly unstable toroidal harmonics in the JOREK simulation broadens the magnetic energy spectrum and ergodises the plasma edge region, resulting in a more localized pressure perturbation. These effects are not observed in VMEC, because closed magnetic flux surfaces are enforced. Despite the sensitivity of JOREK results on the assumed resistivity, saturated states can be found using both approaches that are in reasonable agreement, even for this more advanced case.

Geometric graph-theoretic aspects of quantum stabilizer codes

We propose a systematic procedure for the construction of graphs associated with binary quantum stabilizer codes. The procedure is characterized by means of the following three step process. First, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code. Second, the canonical form of the CWS code is determined and third, the input vertices are attached to the graphs. In order to verify the effectiveness of the procedure, we implement the Gottesman stabilizer code characterized by multi-qubit encoding operators for the resource-efficient error correction of arbitrary single-qubit errors. Finally, the error-correcting capabilities of the Gottesman eight-qubit quantum stabilizer code is verified in graph-theoretic terms as originally advocated by Schlingemann and Werner.

Learning Quantum Circuits of Some T Gates

In this paper, we study the problem of learning an unknown quantum circuit of a certain structure. If the unknown target is an $n$-qubit Clifford circuit, we devise an efficient algorithm to reconstruct its circuit representation by using $O(n^2)$ queries to it. For decades, it has been unknown how to handle circuits beyond the Clifford group since the stabilizer formalism cannot be applied in this case. Herein, we study quantum circuits of $T$-depth one on the computational basis. We show that the output state of a $T$-depth one circuit {\textit{of full $T$-rank}} can be represented by a stabilizer pseudomixture with a specific algebraic structure. Using Pauli and Bell measurements on copies of the output states, we can generate a hypothesis circuit that is equivalent to the unknown target circuit on computational basis states as input. If the number of $T$ gates of the target is of the order $O({{\log n}})$, our algorithm requires $O(n^2)$ queries to it and produces its equivalent circuit representation on the computational basis in time $O(n^3)$. Using further additional $O(4^{3n})$ classical computations, we can derive an exact description of the target for arbitrary input states. Our results greatly extend the previously known facts that stabilizer states can be efficiently identified based on the stabilizer formalism.

Parallel decoding of multiple logical qubits in tensor-network codes

We consider tensor-network stabilizer codes and show that their tensor-network decoder has the property that independent logical qubits can be decoded in parallel. As long as the error rate is below threshold, we show that this parallel decoder is essentially optimal. Holographic codes are interesting examples of tensor-network codes, so we first test out this parallel decoding strategy on the max-rate holographic Steane (heptagon) code. For holographic codes with a constant number of logical qubits, the tensor-network decoder was shown to be efficient with complexity polynomial in $n$, the number of physical qubits. Here we show that, by using the parallel decoding scheme, the complexity is also linear in $k$, the number of logical qubits. We then calculate the bulk threshold (the threshold for logical qubits a fixed distance from the code center) under depolarizing noise for the max-rate holographic Steane code to be $9.4%$. We also introduce some further holographic error-correcting codes and calculate their thresholds under depolarizing noise. One example is based on an 11-qubit code due to Gottesman, which has asymptotic rate 0.114 and threshold 13.8%. Another code we consider is an asymptotically zero-rate holographic code, which performs extremely well under dephasing noise in the $X, Y$, or $Z$ basis with a threshold of $50%$.

Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation. More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion. Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network. The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as non-stabilizer codes over qubits and qudits. For a contractible tensor network, the sequence of contractions also constructs a decoding/encoding circuit. To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct non-trivial codes. These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a 3d stabilizer code, and other stabilizer codes with interesting properties. Surprisingly, we find that the surface code is equivalent to the 2d Bacon-Shor code after "dualizing" its tensor network encoding map.

Theory of synchronisation and pattern formation on time varying networks

Synchronisation and pattern formation have been intensely addressed for systems evolving on static networks. Extending the study to include the inherent ability of the network to adjust over time proved cumbersome and led to conclusions which lack of generality, as relying on peculiar assumptions. Here, the master stability formalism is extended to account, in a thoroughly general prospect, for the additional contributions as stemming from the time evolution of the underlying network. The theory is successfully challenged against two illustrative testbeds, which can be respectively ascribed to synchronisation and Turing settings.

Nature of ideal MHD instabilities as described by multi-region relaxed MHD

In this work, the stepped pressure equilibrium code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502), which computes the equilibria of the multi-region relaxed magnetohydrodynamic energy principle (MRxMHD), has been upgraded to determine the MRxMHD stability in toroidal geometry. A theoretical formalism for SPEC is obtained by relating the second variation of the MRxMHD energy functional to the Hessian matrix, enabling the prediction of magnetohydrodynamic (MHD) linear instabilities. Negative eigenvalues of this matrix imply instability. Further, we demonstrate our method on simplified test scenarios in both tokamak and stellarator magnetic topologies, with a systematic comparison study between the marginal stability prediction of the SPEC with the ideal MHD stability code packages CAS3D and MISHKA-1.

The XYZ2 hexagonal stabilizer code

We consider a topological stabilizer code on a honeycomb grid, the "XYZ2" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six (XYZXYZ) plaquette stabilizers and weight-two (XX) link stabilizers on a planar hexagonal grid composed of 2d2 qubits for code distance d, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure X, Y, or Z noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance d2 only for pure Y noise, here the code distance is 2d2 for both pure Z and pure Y noise. Thresholds for noise with finite Z bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.

Quantum error correction with higher Gottesman-Kitaev-Preskill codes: Minimal measurements and linear optics

We propose two schemes to obtain Gottesman-Kitaev-Preskill (GKP) error syndromes by means of linear optical operations, homodyne measurements and GKP ancillae. This includes showing that for a concatenation of GKP codes with a $[n,k,d]$ stabilizer code only $2n$ measurements are needed in order to obtain the complete syndrome information, significantly reducing the number of measurements in comparison to the canonical concatenated measurement scheme and at the same time generalizing linear-optics-based syndrome detections to higher GKP codes. Furthermore, we analyze the possibility of building the required ancilla states from single-mode states and linear optics. We find that for simple GKP codes this is possible, whereas for concatenations with qubit Calderbank-Shor-Steane (CSS) codes of distance $d\geq3$ it is not. We also consider the canonical concatenated syndrome measurements and propose methods for avoiding crosstalk between ancillae. In addition, we make use of the observation that the concatenation of a GKP code with a stabilizer code forms a lattice in order to see the analog information decoding of such codes from a different perspective allowing for semi-analytic calculations of the logical error rates.

Degenerate local-dimension-invariant stabilizer codes and an alternative bound for the distance preservation condition

One hurdle to performing reliable quantum computations is overcoming noise. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and in particular that degenerate stabilizer codes can have their distance also promised upon sufficiently large local-dimension, as well as a new bound on the local-dimension required to preserve the distance of local-dimension-invariant codes, which is a result which could prove to be useful for error-corrected qudit quantum computers.

Local tensor-network codes

Tensor-network codes enable the construction of large stabilizer codes out of tensors describing smaller stabilizer codes. An application of tensor-network codes was an efficient and exact decoder for holographic codes. Here, we show how to write some topological codes, including the surface code and colour code, as simple tensor-network codes. We also show how to calculate distances of stabilizer codes by contracting a tensor network. The algorithm actually gives more information, including a histogram of all logical coset weights. We prove that this method is efficient in the case of stabilizer codes encoded via local log-depth circuits in one dimension and holographic codes. Using our tensor-network distance calculator, we find a modification of the rotated surface code that has the same distance but fewer minimum-weight logical operators by ‘doping’ the tensor network, i.e., we break the homogeneity of the tensor network by locally replacing tensors. For this example, this corresponds to an improvement in successful error correction of almost 2% against depolarizing noise (in the perfect-measurement setting), but comes at the cost of introducing three higher-weight stabilizers. Our general construction lets us pick a network geometry (e.g., a Euclidean lattice in the case of the surface code), and, using only a small set of seed codes (constituent tensors), build extensive codes with the potential for optimisation.

A Construction of Optimal Nonbinary Pure Quantum Stabilizer Codes

A Construction of Optimal Nonbinary Pure Quantum Stabilizer Codes

Pauli Stabilizer Models of Twisted Quantum Doubles

We construct a Pauli stabilizer model for every two-dimensional Abelian topological order that admits a gapped boundary. Our primary example is a Pauli stabilizer model on four-dimensional qudits that belongs to the double semion (DS) phase of matter. The DS stabilizer Hamiltonian is constructed by condensing an emergent boson in a $\mathbb{Z}_4$ toric code, where the condensation is implemented by making certain two-body measurements. We rigorously verify the topological order of the DS stabilizer model by identifying an explicit finite-depth quantum circuit (with ancillary qubits) that maps its ground state subspace to that of a DS string-net model. We show that the construction of the DS stabilizer Hamiltonian generalizes to all twisted quantum doubles (TQDs) with Abelian anyons. This yields a Pauli stabilizer code on composite-dimensional qudits for each such TQD, implying that the classification of topological Pauli stabilizer codes extends well beyond stacks of toric codes - in fact, exhausting all Abelian anyon theories that admit a gapped boundary. We also demonstrate that symmetry-protected topological phases of matter characterized by type I and type II cocycles can be modeled by Pauli stabilizer Hamiltonians by gauging certain 1-form symmetries of the TQD stabilizer models.

Graph-theoretic approach to quantum error correction

We investigate a novel class of quantum error correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum errors. In this new framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error correcting codes. Most importantly, this framework supports the development of quantum codes that correct against a predetermined set of errors, in contrast to current methods. A heuristic algorithm is presented, providing steps to develop codes that correct against an arbitrary noisy channel. We benchmark the correction capability of reflexive stabilizer codes for the case of single qubit errors by comparison to existing stabilizer codes that are widely used. In addition, we present two instances of optimal encodings: an optimal encoding for fully correlated noise which achieves a higher encoding rate than previously known, and a minimal encoding for single qudit errors on a four-state system.

On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q^2-ary linear codes. This result can be generalized to q^{2 m}-ary linear codes, m > 1. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de, 2020) and new q-ary ones, with q ne 2, improving others in the literature.

Hat problem: a new strategy based on quantum stabilizer codes

Hat problem: a new strategy based on quantum stabilizer codes

Entanglement degree of high-dimensional quantum low-density parity check codes

The important part of quantum information science is quantum error-correction and quantum computation. In thepractical application, some methods of quantum communication, such as the quantum communication based on the atmosphere transmission, require a mix ofmathematics-potentially to solve the problem. The quantum error-correction code is an efficient method to calculate quantum communication. However,since the construction of quantum error-correction codes depends on the degree of quantum entanglement, it is difficult to be widely used in practice.At present, the graph state is a promising solution to reduce the degree of codewords entanglement. But there are still many difficulties in constructingthe high-dimensional graph state. More importantly, the above difficulties can be skillfully solved by the upper and lower bounds of codewords entanglement.Based on the characteristics of the cyclic difference set of stabilizer codes and the combination of U and B of classical low-density parity check (LDPC) codes,a high-dimensional quantum low-density parity check (QLDPC) code is constructed creatively. By calculating the characteristics of the non-$Z$-type generators ofthe new codewords, the minimum number is obtained and the entanglement upper bound of the new code is acquired. And then, the rank of the check matrix of thenew code is calculated as the lower bound of entanglement. When the upper bound and lower bound of entanglement are different, the learning vectorquantization (LVQ) algorithm in machine learning can be used to simultaneously obtain the codewords entanglement and the coding complexity of the code,in which the relationship between them can be derived. In terms of calculating the running speed, compared with the iterative algorithm in the Lagrange multipliermethod, the running speed of LVQ algorithm is increased by $37.68%$. In this paper, a measurement of quantum codewords entanglement is proposed, which supports a helpful role in thedesign of quantum error-correction codes with higher decoding efficiency.

Dynamical signatures of rank-2 U(1) spin liquids

Emergent $U(1)$ gauge theories and artificial photons in frustrated magnets are outstanding examples of many-body collective phenomena. The classical and quantum regimes of these systems provide platforms for classical and quantum spin liquids, and are the subject of current active theoretical and experimental investigations. Recently, realizations of rank-2 $U(1)$ (R2-U1) gauge theories in three-dimensional frustrated magnets have been proposed. Such systems in the quantum regime may lead to the so-called fracton ordered phases---a new class of topological order that has been associated with quantum stabilizer codes and holography. However, there exist few distinguishing characteristics of these states for their detection in real materials. Here we focus on the classical limit and present the dynamical spin structure factor for a R2-U1 spin liquid state on a breathing pyrochlore lattice. Remarkably, we find unique signatures of the R2-U1 state, and we contrast them with the results obtained from a more conventional $U(1)$ spin liquid. These results provide a path of investigation for future inelastic neutron scattering experiments on candidate materials.