Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code

Brown, Benjamin J.; Laubscher, Katharina; Kesselring, Markus S.; Wootton, James R.

doi:10.1103/physrevx.7.021029

Cited by 120 publications

(159 citation statements)

References 94 publications

(195 reference statements)

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“…This type of code deformation does not, in itself, perform logical operations, but can be used to move patches of code or to convert between codes where different gates are transversal [16]. Other code deformation procedures such as moving holes or twists do perform unitary logical Clifford operations [18,23,24]. In the next section, we present another similar procedure which executes a logical measurement.…”

Section: Code Deformation and Lattice Surgery 21 Code Deformationmentioning

confidence: 99%

“…Recently, many different code deformation and lattice surgery techniques have been devised, most of which use tailor-made analysis or decoding techniques, see e.g. [14][15][16][17][18][19][20][21].In this paper, we phrase existing lattice surgery and code deformation protocols as special cases of gauge fixing, showing that the underlying subsystem code dictates the fault-tolerance properties of the protocol. This perspective can simplify the analysis of new measurement-based protocols, provided that they are based on stabilizer codes whose distances can be easily calculated.…”

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Code deformation and lattice surgery are gauge fixing

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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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Section: Code Deformation and Lattice Surgery 21 Code Deformationmentioning

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Code deformation and lattice surgery are gauge fixing

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“…Although the implementation of the Clifford group does not imply the universality of quantum computation (which we would have at the third level of CH), there are important tasks performed by the operations of this group, such as quantum distillation, quantum teleportation and dense coding, for instance. There are other works that propose different techniques for topological coding that also allow the implementation of the Clifford group, as we may see in [21] and [22].…”

Section: Introductionmentioning

confidence: 99%

Construction of color codes from polygons

Silva

Soares

2018

J. Phys. Commun.

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We propose a procedure for constructing new classes of codes, called polygonal color codes. This technique expands the triangular color codes introduced by Bombin and Martin-Delgado in 2007. Triangular color codes, built on a two-dimensional Euclidean surface with three edges, implement the entire Clifford group, but they only encode one qubit. In our case, polygonal color codes are constructed on two-dimensional surfaces with n edges, with n3 , increasing the number of encoded qubits. Considering a surface with boundary and evaluating the generators of the first homology group of this surface, we may increase the dimension of the code without losing the transversal implementation of the Clifford group. In addition, we construct two codes, one with 2 and one with 3 encoded qubits, which exemplify the method and show that we can generate codes with better parameters than the known triangular codes. For one of these constructions, we also give general parameters. OPEN ACCESS RECEIVED

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“…One way to store and manipulate information in these to engineer certain kinds of defects [2,3]. These can be moved around and manipulated in much the same way as particles.…”

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Demonstrating non-Abelian braiding of surface code defects in a five qubit experiment

Wootton

2017

Quantum Sci. Technol.

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Currently, the mainstream approach to quantum computing is through surface codes. One way to store and manipulate quantum information with these to create defects in the codes which can be moved and used as if they were particles. Specifically, they simulate the behaviour of exotic particles known as Majoranas, which are a kind of non-Abelian anyon. By exchanging these particles, important gates for quantum computation can be implemented. Here we investigate the simplest possible exchange operation for two surface code Majoranas. This is found to act non-trivially on only five qubits. The system is then truncated to these five qubits, so that the exchange process can be run on the IBM 5Q processor. The results demonstrate the expected effect of the exchange. This paper has been written in a style that should hopefully be accessible to both professional and amateur scientists.

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Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code

Cited by 120 publications

References 94 publications

Code deformation and lattice surgery are gauge fixing

Code deformation and lattice surgery are gauge fixing

Construction of color codes from polygons

Demonstrating non-Abelian braiding of surface code defects in a five qubit experiment

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