Universality verification for a set of quantum gates

Sawicki, Adam; Mattioli, Lorenzo; Zimborás, Zoltán

doi:10.1103/physreva.105.052602

Cited by 8 publications

(5 citation statements)

References 31 publications

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“…The efficiency of a universal set S (see [8][9][10] for criteria that allow deciding universality) is typically measured by the length of a circuit needed to approximate any quantum transformation with a given precision . The Solovay-Kitaev theorem states that all symmetric universal sets 1 are roughly the same efficient.…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

“…Unitaries constituting δ-approximate t-design form -nets for t d 5/2 and δ 3/2 d d 2 [13]. As a direct consequence of property (1) δ-approximate t-designs find numerous applications throughout quantum information, including randomized benchmarking [16,17], information transmission [18], quantum state discrimination [19], criteria for universality of quantum gates [10] and complexity growth [7,[20][21][22]. It is also known that the constant A(S) from the Solovay-Kitaev theorem is inversely proportional to 1 − δ(ν S ), where δ(ν S ) := sup t δ(ν S , t), whenever δ(ν S ) < 1 [15].…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

“…Lemma 13. For the group SU (d) the constant n 0 is equal to d + 1 and for n ≥ n 0 the formula (10) simplifies to:…”

Section: Frobenius-schur Indicatorsmentioning

confidence: 99%

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Matrix concentration inequalities and efficiency of random universal sets of quantum gates

Dulian

Sawicki

2023

Quantum

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For a random set S⊂U(d) of quantum gates we provide bounds on the probability that S forms a δ-approximate t-design. In particular we have found that for S drawn from an exact t-design the probability that it forms a δ-approximate t-design satisfies the inequality P(δ≥x)≤2Dte−|S|xarctanh(x)(1−x2)|S|/2=O(2Dt(e−x21−x2)|S|), where Dt is a sum over dimensions of unique irreducible representations appearing in the decomposition of U↦U⊗t⊗U¯⊗t. We use our results to show that to obtain a δ-approximate t-design with probability P one needs O(δ−2(tlog⁡(d)−log⁡(1−P))) many random gates. We also analyze how δ concentrates around its expected value Eδ for random S. Our results are valid for both symmetric and non-symmetric sets of gates.

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Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

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Matrix concentration inequalities and efficiency of random universal sets of quantum gates

Dulian

Sawicki

2023

Quantum

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“…We have limited our discussion in this paper to the universality for qubit systems. However, it would be fascinating to explore the usefulness of the GA language in the context of universality problems for higher-dimensional systems, i.e., qudits [78][79][80][81]. Interestingly, for quantum computation by means of qudits [82], a universal set of gates is specified by all one-qudit gates together with any additional entangling two-qudit gate [78] (that is, a gate that does not map separable states onto separable states).…”

Section: Discussionmentioning

confidence: 99%

From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

Cafaro,

Bahreyni,

Rossetti

2024

Symmetry

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Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO3;R and SU2;C depending on the rotor group Spin+3,0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin’s proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin+3,0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods.

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“…In principle, owing to the existence of universal quantum gate sets [1][2][3][4][5] for computation, full-fledged quantum computers [6][7][8][9] that obey the laws of quantum mechanics are promising devices that permit universal fault-tolerant quantum computation [10][11][12][13][14][15][16], with the possibility of surpassing the performances of presently known classical algorithms. In practice, however, numerous challenges remain to be resolved before such a quantum vision can be realized.…”

Section: Introductionmentioning

confidence: 99%

Virtual distillation with noise dilution

Teo¹,

Shin²,

Hyukgun³

et al. 2022

Preprint

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Virtual distillation is an error-mitigation technique that reduces quantum-computation errors without assuming the noise type. In scenarios where the user of a quantum circuit is required to additionally employ peripherals, such as delay lines, that introduce excess noise, we find that the errormitigation performance can be improved if the peripheral, whenever possible, is split across the entire circuit; that is, when the noise channel is uniformly distributed in layers within the circuit. We show that under the multiqubit loss and Pauli noise channels respectively, for a given overall error rate, the average mitigation performance improves monotonically as the noisy peripheral is split (diluted) into more layers, with each layer sandwiched between subcircuits that are sufficiently deep to behave as two-designs. For both channels, analytical and numerical evidence show that second-order distillation is generally sufficient for (near-)optimal mitigation. We propose an application of these findings in designing a quantum-computing cluster that houses realistic noisy intermediate-scale quantum circuits that may be shallow in depth, where measurement detectors are limited and delay lines are necessary to queue output qubits from multiple circuits.

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Universality verification for a set of quantum gates

Cited by 8 publications

References 31 publications

Matrix concentration inequalities and efficiency of random universal sets of quantum gates

Matrix concentration inequalities and efficiency of random universal sets of quantum gates

From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

Virtual distillation with noise dilution

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