Stabilizer Quantum Error-correcting Codes Research Articles

Fault-tolerant double-circular connectivity pattern for quantum stabilizer codes

Recently, the circular connectivity pattern has been presented for a class of stabilizer quantum error correction codes. The circular connectivity pattern for such a class of stabilizer codes can be implemented in a resource-efficient manner using a single ancilla and native two-qubit Controlled-Not-Swap gates (CNS) gates, which may be interesting for demonstrating error-correction codes with superconducting quantum processors. However, one concern is that this scheme is not fault-tolerant. And it might not apply to the Calderbank-Shor-Steane (CSS) codes. In this paper, we present a fault-tolerant version of the circular connectivity pattern, named the double-circular connectivity pattern. This pattern is an implementation for syndrome-measurement circuits with a flagged error correction scheme for stabilizer codes. We illustrate that this pattern is available for Steane code (a CSS code), Laflamme’s five-qubit code, and Shor’s nine-qubit code. For Laflamme’s five-qubit code and Shor’s nine-qubit code, the pattern has the property that it uses only native two-qubit CNS gates, which are more efficient in the superconducting quantum platform.

MDS, Hermitian almost MDS, and Gilbert–Varshamov quantum codes from generalized monomial-Cartesian codes

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in m variables. When m=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=1$$\\end{document} our codes are MDS, and when m=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=2$$\\end{document} and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when m=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=2$$\\end{document} we prove that our codes beat the Gilbert–Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.

Graphical structures for design and verification of quantum error correction

We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the ZX-calculus of quantum observables. The resulting framework leads to a construction for stabilizer codes that allows us to design and verify a broad range of quantum codes based on classical ones, and that gives a means of discovering large classes of codes using both analytical and numerical methods. We focus in particular on the smaller codes that will be the first used by near-term devices. We show how CSS codes form a subset of CPC codes and, more generally, how to compute stabilizers for a CPC code. As an explicit example of this framework, we give a method for turning almost any pair of classical codes into a CPC code. Further, we give a simple technique for machine search which yields thousands of potential codes, and demonstrate its operation for distance 3 and 5 codes. Finally, we use the graphical tools to demonstrate how Clifford computation can be performed within CPC codes. As our framework gives a new tool for constructing small- to medium-sized codes with relatively high code rates, it provides a new source for codes that could be suitable for emerging devices, while its ZX-calculus foundations enable natural integration of error correction with graphical compiler toolchains. It also provides a powerful framework for reasoning about all stabilizer quantum error correction codes of any size.

Realizing a class of stabilizer quantum error correction codes using a single ancilla and circular connectivity

We describe a class of "neighboring-blocks" stabilizer quantum error correction codes and demonstrate that such class of codes can be implemented in a resource-efficient manner using a single ancilla and circular near-neighbor qubit connectivity. We propose an implementation for syndrome-measurement circuits for codes from the class and illustrate its workings for cases of 3-qubit repetition code, Laflamme's 5-qubit code, and Shor's 9-qubit code. For 3-qubit repetition code and Laflamme's 5-qubit code suggested scheme has the property that it uses only native two-qubit CNS gates, which potentially reduces the amount of non-correctable errors due to the shorter gate time. Elements of the scheme can be used to implement surface code with near-neighbour connectivity using single ancilla, as demonstrated in an example. We developed efficient decoding procedures for repetition codes and the Laflamme's 5-qubit code using a minimum weight-perfect matching approach to account for the specific order of measurements in our scheme. The analysis of noise levels for which the scheme could show improvements in the fidelity of a stored logical qubit in the 3-qubit repetition code and Laflamme's 5-qubit code cases is provided. We complement our results by realizing the developed scheme for a 3-qubit code using an IBM quantum processor and the Laflamme's 5-qubit code using the state-vector simulator.

Stabilizer quantum codes defined by trace-depending polynomials

Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace [18]. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in [18] and with excellent parameters. Namely, we are able to provide new binary records according to [21] and non-binary codes improving the ones available in the literature.

Topological graph states and quantum error-correction codes

Deciding if a given family of quantum states is topologically ordered is an important but nontrivial problem in condensed matter physics and quantum information theory. We derive necessary and sufficient conditions for a family of graph states to be in TQO-1, which is a class of quantum error correction code states whose code distance scales macroscopically with the number of physical qubits. Using these criteria, we consider a number of specific graph families, including the star and complete graphs, and the line graphs of complete and completely bipartite graphs, and discuss which are topologically ordered and how to construct the codewords. The formalism is then employed to construct several codes with macroscopic distance, including a three-dimensional topological code generated by local stabilizers that also has a macroscopic number of encoded logical qubits. The results indicate that graph states provide a fruitful approach to the construction and characterization of topological stabilizer quantum error correction codes.

Entanglement-assisted quantum MDS codes from cyclic codes

Entanglement-assisted quantum error-correcting codes are a generalization of standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing pre-shared entanglement between the sender and the receiver. In this paper, we construct seven new families of entanglement-assisted quantum maximum-distance-separable codes from cyclic codes by exploiting less pre-shared entangled states. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.

Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations

Fermion-to-qubit mappings that preserve geometric locality are especially useful for simulating lattice fermion models (e.g., the Hubbard model) on a quantum computer. They avoid the overhead associated with geometric non-local parity terms in mappings such as the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. As a result, they often provide quantum circuits with lower depth and gate complexity. In such encodings, fermionic states are encoded in the common +1 eigenspace of a set of stabilizers, akin to stabilizer quantum error-correcting codes. Here, we discuss several known geometric locality-preserving mappings and their abilities to correct/detect single-qubit errors. We introduce a geometric locality-preserving map, whose stabilizers correspond to products of Majorana operators on closed paths of the fermionic hopping graph. We show that our code, which we refer to as the Majorana loop stabilizer code (MLSC) can correct all single-qubit errors on a 2-dimensional square lattice, while previous geometric locality-preserving codes can only detect single-qubit errors on the same lattice. Compared to existing codes, the MLSC maps the relevant fermionic operators to lower-weight qubit operators despite having higher code distance. Going beyond lattice models, we demonstrate that the MLSC is compatible with state-of-the-art algorithms for simulating quantum chemistry, and can offer those simulations the same error-correction properties without additional asymptotic overhead. These properties make the MLSC a promising candidate for error-mitigated quantum simulations of fermions on near-term devices.

Propagation of generalized Pauli errors in qudit Clifford circuits

It is important for performance studies in quantum technologies to analyze quantum circuits in the presence of noise. We introduce an error probability tensor, a tool to track generalized Pauli error statistics of qudits within quantum circuits composed of qudit Clifford gates. Our framework is compatible with qudit stabilizer quantum error-correcting codes. We show how the error probability tensor can be applied in the most general case, and we demonstrate an error analysis of bipartite qudit repeaters with quantum error correction. We provide an exact analytical solution of the error statistics of the state distributed by such a repeater. For a fixed number of degrees of freedom, we observe that higher-dimensional qudits can outperform qubits in terms of distributed entanglement.

Unitary reconstruction of secret for stabilizer-based quantum secret sharing

We propose a unitary procedure to reconstruct quantum secret for a quantum secret sharing scheme constructed from stabilizer quantum error-correcting codes. Erasure correcting procedures for stabilizer codes need to add missing shares for reconstruction of quantum secret, while unitary reconstruction procedures for certain class of quantum secret sharing are known to work without adding missing shares. The proposed procedure also works without adding missing shares.

Undetermined states: how to find them and their applications

We investigate the undetermined sets consisting of two-level, multi-partite pure quantum states, whose reduced density matrices give absolutely no information of their original states. Two approached of finding these quantum states are proposed. One is to establish the relation between codewords of the stabilizer quantum error correction codes (SQECCs) and the undetermined states. The other is to study the local complementation rules of the graph states. As an application, the undetermined states can be exploited in the quantum secret sharing scheme. The security is guaranteed by their undetermineness.

Stabilizer quantum error correction with quantum bus computation

In this paper we investigate stabilizer quantum error correction codes using controlled phase rotations of strong coherent probe states. We explicitly describe two methods to measure the Pauli operators that generate the stabilizer group of a quantum code. First, we show how to measure a Pauli operator acting on physical qubits using a single coherent state with large average photon number, displacement operations, and photon detection. Second, we show how to measure the stabilizer operators fault-tolerantly by the deterministic preparation of coherent quantum superposition (``cat'') states along with one-bit teleportations between a qubitlike encoding of coherent states and physical qubits.