Exact and approximate unitary 2-designs and their application to fidelity estimation

Dankert, Christoph; Cleve, Richard; Emerson, Joseph; Livine, Etera R.

doi:10.1103/physreva.80.012304

Cited by 631 publications

(841 citation statements)

References 24 publications

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“…Clif n plays an important role in many areas of quantum information such as universality [45], stabilizer code theory/fault-tolerance [46] and noise estimation [17].…”

Section: B the Clifford Group And T-designsmentioning

confidence: 99%

“…(2.23) being satisfied for any quantum channel Λ and any state ρ [17]. Since a uniform probability distribution on Clif n forms a 2-design, if Clif n = {C j : j ∈ K = {1, ..., |Clif n |}} then,…”

Section: Definition 1 Unitary T-designmentioning

confidence: 99%

“…A complete characterization of the noise is useful because it allows for the determination of good error-correction schemes, and thus the possibility of reliable transmission of quantum information. Since complete process tomography is infeasible for large systems, there is growing interest in scalable methods for partially characterizing the noise affecting a quantum system [17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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Characterizing quantum gates via randomized benchmarking

Gambetta²,

2012

Self Cite

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We describe and expand upon the scalable randomized benchmarking protocol proposed in Phys. Rev. Lett. 106, 180504 (2011) which provides a method for benchmarking quantum gates and estimating the gate-dependence of the noise. The protocol allows the noise to have weak time and gate-dependence, and we provide a sufficient condition for the applicability of the protocol in terms of the average variation of the noise. We discuss how state preparation and measurement errors are taken into account and provide a complete proof of the scalability of the protocol. We establish a connection in special cases between the error rate provided by this protocol and the error strength measured using the diamond norm distance.

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“…Clif n plays an important role in many areas of quantum information such as universality [45], stabilizer code theory/fault-tolerance [46] and noise estimation [17].…”

Section: B the Clifford Group And T-designsmentioning

confidence: 99%

Section: Definition 1 Unitary T-designmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characterizing quantum gates via randomized benchmarking

Gambetta²,

2012

Self Cite

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show abstract

“…The entanglement entropy, and even its time dependence, is also beginning to be experimentally measurable in cold atom systems [32][33][34]. In a very different context, black holes have motivated studies of how fast quantum systems can scramble information by dynamically generating entanglement [35][36][37][38]. Simple quantum circuitsquantum evolutions in discrete time-serve as useful toy models for entanglement growth and scrambling [39][40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the "entanglement tsunami" in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated KardarParisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like ðtimeÞ 1=3 and are spatially correlated over a distance ∝ ðtimeÞ 2=3 . We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a "minimal cut" picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the "velocity" of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the wellstudied problem of pinning of a membrane or domain wall by disorder.

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“…Some constructions of 2-designs were given by Dankert, Cleve, Emerson and Livine [6] and Gross, Audenaert and Eisert [14]. The argument of Seymour and Zaslavsky [21] shows that U(d) t-designs exist for every t and d [19,Corollary 5.3], and the result of the previous section bounds the size of the smallest weighted t-design to O(d 4t ).…”

Section: Constructionsmentioning

confidence: 95%

Unitary designs and codes

Roy

Scott

2009

Des. Codes Cryptogr.

100

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A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code -a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values -and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary tdesign in U(d), and we catalogue some t-designs that arise from finite groups. *

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Exact and approximate unitary 2-designs and their application to fidelity estimation

Cited by 631 publications

References 24 publications

Characterizing quantum gates via randomized benchmarking

Characterizing quantum gates via randomized benchmarking

Quantum Entanglement Growth under Random Unitary Dynamics

Unitary designs and codes

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