Fault-tolerant error correction with the gauge color code

Brown, Benjamin J.; Nickerson, Naomi; Browne, Dan E.

doi:10.1038/ncomms12302

Cited by 98 publications

(95 citation statements)

References 54 publications

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“…He argued that we need to measure each face operator of the gauge color code only once to obtain reliable fault-tolerant syndrome information. This phenomenon, coined single-shot error correction, has been demonstrated numerically by Brown, Nickerson, and Browne (2016). This differs from the well-studied case of twodimensional stabilizer models where syndrome information is read out using an unreliable measurement apparatus.…”

Section: The Gauge Color Codementioning

confidence: 85%

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Quantum memories at finite temperature

et al. 2016

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To use quantum systems for technological applications one first needs to preserve their coherence for macroscopic time scales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. The state-of-the-art developments in this field are reviewed in an informative and pedagogical way. The main principles of self-correcting quantum memories are given and several milestone examples from the literature of two-, three-and higher-dimensional quantum memories are analyzed.

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Section: The Gauge Color Codementioning

confidence: 85%

“…22. Another appropriate lattice geometry is given by Kim (2011); see also Brown, Nickerson, and Browne (2016). Importantly, the lattice is four valent and four colorable, i.e., the lattice is such that we can assign to each cell one of four colors in such a way that no two neighboring cells are of the same color.…”

Section: The Gauge Color Codementioning

confidence: 99%

Quantum memories at finite temperature

et al. 2016

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“…One would expect that in a fully fault-tolerant simulation the gate error threshold for the μ-dimensional color codes would be prohibitively low, since the stabilizers defined on the μ-cells would require circuits of many time steps to measure. There is hope, however, that by switching to the corresponding gauge color code the measurements require fewer gates and therefore the threshold may recover [52]. The development of efficient decoders for qudit gauge color codes is therefore of significant interest.…”

Section: Discussionmentioning

confidence: 99%

“…For the case of gauge color codes, very recently a harddecision RG decoder [52] was implemented for the qubit 3D construction. As with the color code, we expect a generalization of such an algorithm to gauge color codes in arbitrary spatial and qudit dimensions to be straightforward.…”

Section: Error Detectionmentioning

confidence: 99%

Qudit color codes and gauge color codes in all spatial dimensions

et al. 2015

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Two-level quantum systems, qubits, are not the only basis for quantum computation. Advantages exist in using qudits, d-level quantum systems, as the basic carrier of quantum information. We show that color codes, a class of topological quantum codes with remarkable transversality properties, can be generalized to the qudit paradigm. In recent developments it was found that in three spatial dimensions a qubit color code can support a transversal non-Clifford gate and that in higher spatial dimensions additional non-Clifford gates can be found, saturating Bravyi and König's bound [S. Bravyi and R. König, Phys. Rev. Lett. 111, 170502 (2013)]. Furthermore, by using gauge fixing techniques, an effective set of Clifford gates can be achieved, removing the need for state distillation. We show that the qudit color code can support the qudit analogs of these gates and also show that in higher spatial dimensions a color code can support a phase gate from higher levels of the Clifford hierarchy that can be proven to saturate Bravyi and König's bound in all but a finite number of special cases. The methodology used is a generalization of Bravyi and Haah's method of triorthogonal matrices [S. Bravyi and J. Haah, Phys. Rev. A 86, 052329 (2012)], which may be of independent interest. For completeness, we show explicitly that the qudit color codes generalize to gauge color codes and share many of the favorable properties of their qubit counterparts.

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“…Most leading laboratories are following designs [3][4][5] within this paradigm of distill-thensynthesize combined with surface codes. While alternative ideas to magic state distillation exist [6][7][8][9], so far they lack the appealing high tolerance to noise [10,11]. We propose a framework where both distillation and synthesis occur simultaneously, which we call synthillation.…”

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Unifying Gate Synthesis and Magic State Distillation

Campbell

Howard

2017

Phys. Rev. Lett.

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The leading paradigm for performing computation on quantum memories can be encapsulated as distill-thensynthesize. Initially, one performs several rounds of distillation to create high-fidelity magic states that provide one good T gate, an essential quantum logic gate. Subsequently, gate synthesis intersperses many T gates with Clifford gates to realise a desired circuit. We introduce a unified framework that implements one round of distillation and multi-qubit gate synthesis in a single step. Typically, our method uses the same number of T -gates as conventional synthesis, but with the added benefit of quadratic error suppression. Because of this, one less round of magic state distillation needs to be performed, leading to significant resource savings.Development of quantum computers has intensified, spurred on by the prospect that fully fault-tolerant devices are within reach. A major impetus has been new theoretical advances showing practical designs of fault-tolerant devices can tolerate up to one percent noise [1]. The topological surface code or toric code is the most widely known breakthrough, which allows for a robust storage of quantum information. Augmenting the surface code from a static memory to a computer requires additional information processing gadgets. Fault-tolerant information processing can be achieved by a two-step process. In the first step, logical qubits are distilled from noisy resources into high-fidelity magic states [2]. Each magic state can provide a fault-tolerant T -gate, also known as a π/8 phase gate. In the second step, gate-synthesis techniques decompose any desired unitary into a sequence of many T -gates interspersed with Clifford gates. This approach to processing quantum information can be paraphrased as distill-then-synthesize. Most leading laboratories are following designs [3][4][5] within this paradigm of distill-thensynthesize combined with surface codes. While alternative ideas to magic state distillation exist [6][7][8][9], so far they lack the appealing high tolerance to noise [10,11]. We propose a framework where both distillation and synthesis occur simultaneously, which we call synthillation.Fault-tolerance protocols come with a price-tag, an overhead of extra qubits. Consequently, genuinely useful applications may need millions or billions of physical qubits. Improved protocols for magic state distillation [12][13][14] and gate synthesis [15][16][17][18][19][20][21] have reduced resource overheads, but the cost remains formidable and further overhead reduction is extremely valuable. Notable is the Bravyi-Haah magic state distillation (BHMSD) protocol [13] that converts 3k + 8 magic states into k magic states with quadratic error suppression. For large computations, with between 10 10 and 10 15 logical operations, the required precision can be reached by concatenating BHMSD two or three times, assuming an initial physical error rate of order ∼ 0.1%. Multilevel distillation is an effective tool when many rounds are required [14]. Gate synthesis has advanced on tw...

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Fault-tolerant error correction with the gauge color code

Cited by 98 publications

References 54 publications

Quantum memories at finite temperature

Quantum memories at finite temperature

Qudit color codes and gauge color codes in all spatial dimensions

Unifying Gate Synthesis and Magic State Distillation

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