Complexity of quantum circuits via sensitivity, magic, and coherence

Bu, Kaifeng; Garcia, Roy J.; Jaffe, Arthur; Koh, Dax Enshan; Li, Lü

doi:10.48550/arxiv.2204.12051

Cited by 6 publications

(10 citation statements)

References 77 publications

(96 reference statements)

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“…N (3) h q. systems n 0 Gaussian vars Smooth M (4) N (4) h q. systems n 0 Gaussian vars Multilinearization bounded q. systems bounded Gaussian vars bounded deg & multilinear Lemma 5.21 Lemma 5.21 M (5) N (5) h q. systems n 0 n 1 Gaussian vars Invariance principle bounded q. systems Lemma 5.13 Lemma 5.13 M (6) N (6) h + n 0 n 1 q. systems Rounding quantum operations Lemma 5.24 Lemma 5.24 J ((Φ Alice ) * ) and J ((Φ Bob ) * ) on the input spaces. Note that both Choi representations are positive operators.…”

Section: Smoothingmentioning

confidence: 99%

“…In this step we substitute the Gaussian variables with properly chosen basis elements, to get operators M (6) and N (6) , which have a bounded number of quantum subsystems. Again, we need to apply a quantum invariance principle to ensure that M (6) and N (6) are close to positive operators.…”

Section: Invariance To Operatorsmentioning

confidence: 99%

“…We now have operators M (6) and N (6) that are close to positive operators. The last thing to do is to round them to the Choi representations of the adjoints of some quantum operations.…”

Section: Roundingmentioning

confidence: 99%

“…Meanwhile, Fourier analysis in quantum settings has found applications in various topics, such as quantum communication complexity [2], circuit complexity [6], property testing of unitary operators [40], learning quantum juntas [9], learning quantum dynamics [37], etc. It is fascinating to see more applications of this growing field in quantum information and quantum computation.…”

Section: Open Problemsmentioning

confidence: 99%

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Nonlocal Games with Noisy Maximally Entangled States are Decidable

Qin¹,

Yao²

2021

SIAM J. Comput.

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This paper considers the decidability of fully quantum nonlocal games with noisy maximally entangled states. Fully quantum nonlocal games are a generalization of nonlocal games, where both questions and answers are quantum and the referee performs a binary POVM measurement to decide whether they win the game after receiving the quantum answers from the players. The quantum value of a fully quantum nonlocal game is the supremum of the probability that they win the game, where the supremum is taken over all the possible entangled states shared between the players and all the valid quantum operations performed by the players. The seminal work MIP * = RE [23,24] implies that it is undecidable to approximate the quantum value of a fully nonlocal game. This still holds even if the players are only allowed to share (arbitrarily many copies of) maximally entangled states. This paper investigates the case that the shared maximally entangled states are noisy. We prove that there is a computable upper bound on the copies of noisy maximally entangled states for the players to win a fully quantum nonlocal game with a probability arbitrarily close to the quantum value. This implies that it is decidable to approximate the quantum values of these games. Hence, the hardness of approximating the quantum value of a fully quantum nonlocal game is not robust against the noise in the shared states.This paper is built on the framework for the decidability of non-interactive simulations of joint distributions [17,12,16] and generalizes the analogous result for nonlocal games in [35]. We extend the theory of Fourier analysis to the space of super-operators and prove several key results including an invariance principle and a dimension reduction for super-operators. These results are interesting in their own right and are believed to have further applications.

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Section: Smoothingmentioning

confidence: 99%

Section: Invariance To Operatorsmentioning

confidence: 99%

“…We now have operators M (6) and N (6) that are close to positive operators. The last thing to do is to round them to the Choi representations of the adjoints of some quantum operations.…”

Section: Roundingmentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

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Nonlocal Games with Noisy Maximally Entangled States are Decidable

Qin¹,

Yao²

2021

SIAM J. Comput.

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“…Quantum complexity of a unitary transformation is then defined as the smallest number of logic gates needed to implement the unitary transformation (c.f. [1,2,3,5,6,7,8,9,10]). In this article, we study the quantum complexity of permutations, a special type of unitary transformations.…”

Section: Introductionmentioning

confidence: 99%

Quantum Complexity of Permutations

Yu¹

2022

Preprint

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Let S n be the symmetric group of all permutations of {1, • • • , n} with two generators: the transposition switching 1 with 2 and the cyclic permutation sending k to k + 1 for 1 ≤ k ≤ n − 1 and n to 1 (denoted by σ and τ ). In this article, we study quantum complexity of permutations in S n using {σ, τ, τ −1 } as logic gates. We give an explicit construction of permutations in S n with quadratic quantum complexity lower bound n 2 −2n−7 4. We also prove that all permutations in S n have quadratic quantum complexity upper bound 3(n − 1) 2 . Finally, we show that almost all permutations in S n have quadratic quantum complexity lower bound when n → ∞.

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Circuit complexity of quantum access models for encoding classical data

Zhang,

Yuan

2024

npj Quantum Inf

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How to efficiently encode classical data is a fundamental task in quantum computing. While many existing works treat classical data encoding as a black box in oracle-based quantum algorithms, their explicit constructions are crucial for the efficiency of practical algorithm implementations. Here, we unveil the mystery of the classical data encoding black box and study the Clifford + T complexity in constructing several typical quantum access models. For general matrices (even including sparse ones), we prove that sparse-access input models and block-encoding both require nearly linear circuit complexities relative to the matrix dimension. We also give construction protocols achieving near-optimal gate complexities. On the other hand, the construction becomes efficient with respect to the data qubit when the matrix is a linear combination of polynomial terms of efficiently implementable unitaries. As a typical example, we propose improved block-encoding when these unitaries are Pauli strings. Our protocols are built upon improved quantum state preparation and a select oracle for Pauli strings, which hold independent values. Our access model constructions provide considerable flexibility, allowing for tunable ancillary qubit numbers and offering corresponding space-time trade-offs.

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Complexity of quantum circuits via sensitivity, magic, and coherence

Cited by 6 publications

References 77 publications

Nonlocal Games with Noisy Maximally Entangled States are Decidable

Nonlocal Games with Noisy Maximally Entangled States are Decidable

Quantum Complexity of Permutations

Circuit complexity of quantum access models for encoding classical data

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