Rewiring stabilizer codes

Colladay, Kristina; Mueller, Erich J.

doi:10.1088/1367-2630/aad8dd

Cited by 14 publications

(14 citation statements)

References 29 publications

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“…j=1 , of the cosets of Z(S) in G N (18) and of the cosets of S in Z(S) (20), are often referred to in the literature as pure errors and logical operators 5 , respectively [79]- [84]. Expressions (19) and ( 22) explain why the terms "pure error" and "logical operator" are chosen for {T i } 2 N −k i=1 and {L j } 2 2k j=1 .…”

Section: Pure Errors and Logical Operatorsmentioning

confidence: 99%

Degeneracy and Its Impact on the Decoding of Sparse Quantum Codes

et al. 2021

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Section: Pure Errors and Logical Operatorsmentioning

confidence: 99%

Degeneracy and Its Impact on the Decoding of Sparse Quantum Codes

et al. 2021

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“…This provides a type theory for stabilizer codes that includes encoding and decoding [3], and syndrome extraction [19] circuits. Potential other applications could include analyzing circuits that implement non-Clifford gates on stabilizer codes [13,21] and circuits that fault-tolerantly switch between codes [4].…”

Section: Applications and Future Workmentioning

confidence: 99%

Gottesman Types for Quantum Programs

Rand

Sundaram

Singhal

et al. 2021

Electron. Proc. Theor. Comput. Sci.

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The Heisenberg representation of quantum operators provides a powerful technique for reasoning about quantum circuits, albeit those restricted to the common (non-universal) Clifford set H, S and CNOT . The Gottesman-Knill theorem showed that we can use this representation to efficiently simulate Clifford circuits. We show that Gottesman's semantics for quantum programs can be treated as a type system, allowing us to efficiently characterize a common subset of quantum programs. We also show that it can be extended beyond the Clifford set to partially characterize a broad range of programs. We apply these types to reason about separable states and the superdense coding algorithm.

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“…Figure B2. Comparison between Steane-to-Reed-Muller conversion schemes from [31] (top) and [32] (bottom). Red and green tinting match figure B1, blue tinting indicates an X operator supported on the vertices of the tinted face or volume.…”

Section: Appendix C Disparity In Error Rates Of Cnot Gatesmentioning

confidence: 99%

“…The second scheme, from Colladay and Mueller [32], is not based on gauge fixing, and begins with the eight qubits needed for conversion initialized in the state ñ Ä |0 8 . This ensures that the initial checks anticommute with Table A1.…”

Section: Appendix a Algebraic Proof Of The Correctness Of The Merge mentioning

confidence: 99%

Code deformation and lattice surgery are gauge fixing

et al. 2019

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The large-scale execution of quantum algorithms requires basic quantum operations to be implemented fault-tolerantly. The most popular technique for accomplishing this, using the devices that can be realized in the near term, uses stabilizer codes which can be embedded in a planar layout. The set of fault-tolerant operations which can be executed in these systems using unitary gates is typically very limited. This has driven the development of measurement-based schemes for performing logical operations in these codes, known as lattice surgery and code deformation. In parallel, gauge fixing has emerged as a measurement-based method for performing universal gate sets in subsystem stabilizer codes. In this work, we show that lattice surgery and code deformation can be expressed as special cases of gauge fixing, permitting a simple and rigorous test for fault-tolerance together with simple guiding principles for the implementation of these operations. We demonstrate the accuracy of this method numerically with examples based on the surface code, some of which are novel.Several such families of quantum error-correcting codes have been developed, including concatenated codes [5,6], subsystem codes such as Bacon-Shor codes [7], and 2D topological codes. The most prominent 2D topological codes are surface codes [8] derived from Kitaev's toric code [9], which we will focus on in the remainder of this manuscript. 2D topological codes can be implemented using entangling gates which are local in two dimensions, allowing fault-tolerance in near-term devices which have limited connectivity. In addition, 2D topological codes generally have high fault-tolerant memory thresholds, with the surface code having the highest at ∼1% [10].These advantages come at a cost, however. While other 2D topological codes permit logical single-qubit Clifford operations to be implemented transversally, the surface code does not. In addition, the constraint that computation be carried out in a single plane does not permit two-qubit physical gates to be carried out between physical qubits in different code blocks, precluding the two-qubit gates which, in principle, can be carried out transversally.These two restrictions have led to the design of measurement-based protocols for performing single-and two-qubit logical gates by making gradual changes to the underlying stabilizer code. Measurement-based protocols that implement single-qubit gates are typically called code deformation [11], and protocols that involve multiple logical qubits are usually called lattice surgery [12]. A separate measurement-based technique, called gauge fixing [13], can be applied to subsystem codes, which have operators which can be added to or removed from the stabilizer group as desired, the so-called gauge operators. During gauge fixing, the stabilizer generators of the subsystem code remain unchanged, and can be used to detect and correct errors; so decoding is unaffected by gauge fixing. This is in contrast to code deformation and lattice surgery, where it is not a pri...

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Rewiring stabilizer codes

Cited by 14 publications

References 29 publications

Degeneracy and Its Impact on the Decoding of Sparse Quantum Codes

Degeneracy and Its Impact on the Decoding of Sparse Quantum Codes

Gottesman Types for Quantum Programs

Code deformation and lattice surgery are gauge fixing

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