Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions

Haferkamp, Jonas; Hunter-Jones, Nicholas

doi:10.1103/physreva.104.022417

Cited by 13 publications

(16 citation statements)

References 31 publications

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“…An important example of approximate k-designs are the 1D architecture random quantum circuits formed from arbitrary universal gates that randomly couple neighbouring qubits. Those easy to implement circuits approximate k-designs efficiently with the number of qubits N [54][55][56]. Specifically, δ-approximate 4-designs are generated by local random quantum circuits of depth O((N (N + log(1/δ)) and by the random brickwork architecture in depth O(N + log(1/δ)), with moderate numerical constants (see Table 1 of [56] for the exact scaling).…”

Section: Exact and Approximate Unitary K-designsmentioning

confidence: 99%

“…Remark 7. We note that random quantum circuits in the 1D architecture formed from arbitrary universal gates that randomly couple neighbouring qubits, generate approximate k-designs efficiently with the number of qubits N [54][55][56]. Specifically, δapproximate 4-designs are generated by the 1D random brickwork architecture in depth O(N + log(1/δ)), with moderate numerical constants [56].…”

Section: A Quantum Statesmentioning

confidence: 99%

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Exploring Quantum Average-Case Distances: proofs, properties, and examples

Maciejewski¹,

Puchała²,

Oszmaniec³

2021

Preprint

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In this work, we perform an in-depth study of recently introduced average-case quantum distances . The average-case distances approximate, up to relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantify average-case statistical distinguishability via random quantum circuits.We prove that once a family of random circuits forms an δ-approximate 4-design, with δ = o(d −8 ), then the average-case distances can be approximated by simple explicit functions that can be expressed via simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed -no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all of the distances utilize the Hilbert-Schmidt (HS) norm in some way. This gives the HS norm an operational interpretation that it did not possess before. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them we provide an example that saturates the bound. Specifically, we show that for each dimension d this ratio is at most d 1 2 , d, d 3 2 for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically. Furthermore, we perform an extensive numerical study of systems up to 10 qubits where circuits have a form of variational ansätze with randomly chosen parameters. Despite the fact that such circuits are not known to form unitary designs, our results suggest that average-case distances are more suitable than the conventional distances for studying the performance of NISQ devices in such settings.

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Section: Exact and Approximate Unitary K-designsmentioning

confidence: 99%

Section: A Quantum Statesmentioning

confidence: 99%

Exploring Quantum Average-Case Distances: proofs, properties, and examples

Maciejewski¹,

Puchała²,

Oszmaniec³

2021

Preprint

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“…Indeed, Ref. [BCHJ + 21] implies such a linear lower bound in the limit of large local dimension [HJ19] (it was later shown that a scaling of ∼ 6t 2 for the local dimension is sufficient [HHJ21]). More recently, a variant of the Brown-Susskind conjecture for random quantum circuits was proven for the special case of the exact circuit complexity [HFK + 21].…”

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confidence: 99%

Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$

Haferkamp

2022

Preprint

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The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary t-designs. Unitary t-designs are probability distributions that mimic Haar randomness up to tth moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits in a brickwork architecture of depth O(nt 10.5 ) are approximate unitary t-design. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by Ω(n −1 t −9.5 ). We improve this lower bound to Ω(n −1 t −4−o(1) ). A direct consequence of this scaling is that random quantum circuits generate approximate unitary t-designs in depth O(nt 5+o(1) ). Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove a near optimal convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries in Wasserstein distance.

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“…Importantly, random quantum circuits in the 1D architecture formed from arbi-trary universal gates that randomly couple neighbouring qubits, generate approximate k-designs efficiently with the number of qubits N [33][34][35][36]. Specifically, δ-approximate 4-designs are generated by the 1D random brickwork architecture in depth O(N + log(1/δ)), with moderate numerical constants [36].…”

mentioning

confidence: 99%

“…See technical version of the paper [29] for details and proofs of various properties of average-case distances. Fourth, while it may seem that condition of being (approximate) 4-design is quite stringent, from recent paper [36] it follows that ensembles of quantum circuits required by Theorems 1-3 can be realised by random circuits in the 1D brickwork architecture in depth O(N ) (with moderate prefactors) [36]. Finally, we expect that our average-case distances will more accurately capture behavior of errors in performance of quantum objects in generic moderate size quantum algorithms (note that many architectures of variational circuits used in NISQ algorithms are expected to exhibit, on average, design-like behavior [14]).…”

mentioning

confidence: 99%

Operational Quantum Average-Case Distances

Maciejewski¹,

Puchała²,

Oszmaniec³

2021

Preprint

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We introduce operational distance measures between quantum states, measurements, and channels based on their average-case distinguishability. To this end, we analyze the average Total Variation Distance (TVD) between statistics of quantum protocols in which quantum objects are intertwined with random circuits and subsequently measured in the computational basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects, which we call average-case distances. The so-defined distances capture average-case distinguishability via moderate-depth random quantum circuits and satisfy many natural properties. We apply them to analyze effects of noise in quantum advantage experiments and in the context of efficient discrimination of high-dimensional quantum states and channels without quantum memory. Furthermore, based on analytical and numerical examples, we argue that average-case distances are better suited for assessing the quality of NISQ devices than conventional distance measures such as trace distance and the diamond norm.Introduction. In the era of Noisy Intermediate Scale Quantum (NISQ) devices [1], it is instrumental to have figures of merit that quantify how close two quantum protocols are. The distance measures commonly used for this purpose, for example, in the context of quantum error-correction [2], such as trace distance or diamond norm, have an operational interpretation in terms of optimal statistical distinguishability between two quantum states, measurements, or channels [3][4][5][6]. While it is natural to consider the optimal protocols when one wishes to distinguish between two objects, alas, in reality, such protocols might be not practical. For example, in general, they require high-depth, complicated quantum circuits [7]. From a complementary perspective, quantum distances are often used to compare an ideal implementation (of a state, measurement, or channel) with its noisy experimental version. In this context, using the distances based on optimal distinguishability gives information about the worst-case performance of a device in question. This may be impractical as well -it is not expected that the performance of typical experiments on a quantum device will be comparable to the worst-case scenario.

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Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions

Cited by 13 publications

References 31 publications

Exploring Quantum Average-Case Distances: proofs, properties, and examples

Exploring Quantum Average-Case Distances: proofs, properties, and examples

Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$

Operational Quantum Average-Case Distances

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