Restrictions on Transversal Encoded Quantum Gate Sets

Eastin, Bryan; Knill, Emanuel

doi:10.1103/physrevlett.102.110502

Cited by 324 publications

(375 citation statements)

References 9 publications

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“…Moreover, this result is restricted to models that give rise to fault-tolerant quantum computation by transversal gates. Indeed, quantum coding theory has shown that universal transversal operations are known to be incompatible with stabilizer error correction (Eastin and Knill, 2009;Zeng, Cross, and Chuang, 2011; Anderson and Jochym-O'Connor, 2014). The reader may question this remark as to how the six-dimensional color code achieves a universal set of operations given known restrictions on universal transversal gate sets.…”

Section: Thermally Stable Quantum Computationmentioning

confidence: 99%

“…As an aside remark, the nontrivial braiding statistics between anyons can be obtained from the commutation relations of logical operators (Einarsson, 1990) The ground space of the toric code is naturally described with a basis labeled by anyonic charges a, wrapping around a nontrivial cycle of the torus and then annihilating with its antiparticle, such as that shown in red. (b) Two anyonic excitations created that can propagate at no energy cost to affect the ground space of the system.…”

Section: E Topological Order and Anyons In The Toric Codementioning

confidence: 99%

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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: Thermally Stable Quantum Computationmentioning

confidence: 99%

Section: E Topological Order and Anyons In The Toric Codementioning

confidence: 99%

Quantum memories at finite temperature

et al. 2016

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“…Classification of fault-tolerant implementable logical gates in topological quantum error-correcting codes is an important stepping stone toward far-reaching goal of universal quantum computation [1][2][3][4]. Characterization of logical operators is also essential in understanding braiding and fusion rules of anyonic excitations arising in topologically ordered systems [5,6].…”

Section: Introductionmentioning

confidence: 99%

Topological color code and symmetry-protected topological phases

Yoshida

2015

Phys. Rev. B

133

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We study (d − 1)-dimensional excitations in the d-dimensional color code that are created by transversal application of the R d phase operators on connected subregions of qubits. We find that such excitations are the superpositions of electric charges and can be characterized by the fixed-point wave functions of (d − 1)-dimensional bosonic symmetry-protected topological (SPT) phases with (Z 2 ) ⊗d symmetry. While these SPT excitations are localized on (d − 1)-dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the nontriviality of emerging SPT wave functions. Moreover, these SPT excitations can be physically realized as transparent gapped domain walls which exchange excitations in the color code. Namely, in the three-dimensional color code, the domain wall, associated with the transversal R 3 operator, exchanges a magnetic flux and a composite of a magnetic flux and the looplike SPT excitation, revealing rich possibilities of boundaries in higher-dimensional TQFTs. We also find that magnetic fluxes and the looplike SPT excitations exhibit nontrivial three-loop braiding statistics in three dimensions as a result of the fact that the R 3 phase operator belongs to the third level of the Clifford hierarchy. We believe that the connection between SPT excitations, fault-tolerant logical gates and gapped domain walls, established in this paper, can be generalized to a large class of topological quantum codes and TQFTs.

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“…Stabilizer codes only allow coherent implementation of a limited group of fault tolerant gates, the so-called transversal gates. Unfortunately, recent research has shown that no stabilizer code can both protect against generic errors and offer a universal set of transversal gates [5]. Similarly, topologically protected groups of gates, implemented by braiding anyons, are not universal for many species of anyons [6][7][8].…”

mentioning

confidence: 99%

Catalysis and activation of magic states in fault-tolerant architectures

Campbell

2011

Phys. Rev. A

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In many architectures for fault tolerant quantum computing universality is achieved by a combination of Clifford group unitary operators and preparation of suitable nonstabilizer states, the so-called magic states. Universality is possible even for some fairly noisy nonstabilizer states, as distillation can convert many noisy copies into fewer purer magic states. Here we propose novel protocols that exploit multiple species of magic states in surprising ways. These protocols provide examples of previously unobserved phenomena that are analogous to catalysis and activation well known in entanglement theory.PACS numbers: 03.67.Pp Quantum computers are capable of executing algorithms whilst tolerating modest rates of faults or errors. Stabilizer codes encode information in subspaces of larger Hilbert spaces and allow a proportion of errors to be actively detected and corrected [1]. Whereas some anyonic systems with topologically protected ground states provide a passive method of safely storing quantum information [2]. Research into anyonic systems has been stimulated by the recent discovery of alloys that are topological insulators [3,4], opening up a variety of readily available systems that may be suitable for anyonic quantum computing.However, fault tolerant quantum computing is not just about archiving quantum information, but also processing the information whilst stored in its protected form. However, by employing stabilizer codes and topological systems we restrict how the quantum information may be fault-tolerantly manipulated. Stabilizer codes only allow coherent implementation of a limited group of fault tolerant gates, the so-called transversal gates. Unfortunately, recent research has shown that no stabilizer code can both protect against generic errors and offer a universal set of transversal gates [5]. Similarly, topologically protected groups of gates, implemented by braiding anyons, are not universal for many species of anyons [6][7][8]. Theoretically, some exotic anyons do offer universal topologically protected gates, but these are more physically speculative [9]. Consequently, an alternative route to universal and fault tolerant quantum computing must be sought out.This obstacle is overcome by gate injection techniques. A suitable resource state is identified, and through fault tolerant gates and measurements, this resource is consumed in exchange for a new fault-tolerant unitary operator that promotes the group of gates to full universality. For both stabilizer codes and anyonic systems, the manifestly fault tolerant gates are often contained within the Clifford group, the group of unitary operators that conjugate the Pauli operators. What resource states might promote the Clifford group to universality? Since the Clifford group maps stabilizer states -eigenstates of Pauli operators -to other stabilizer states, and such evolutions are efficiently classically simulable [10], we know that stabilizer states fail to provide universality. However, numer- * Electronic address: earltcampbell@gmail.com...

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Restrictions on Transversal Encoded Quantum Gate Sets

Cited by 324 publications

References 9 publications

Quantum memories at finite temperature

Quantum memories at finite temperature

Topological color code and symmetry-protected topological phases

Catalysis and activation of magic states in fault-tolerant architectures

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