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Qudit quantum Programming with Projective Cliffords

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arXiv 2024年

作者： Paykin, Jennifer Winnick, Sam Intel Labs Intel Corporation United States Institute for Quantum Computing Department of Combinatorics & Optimization University of Waterloo Canada

This paper introduces a novel abstraction for programming quantum operations, specifically projective Cliffords, as functions over the qudit Pauli group. We define a categorical semantics for projective Cliffords based on Pauli encodings in terms of Zd-linear maps. We then introduce a type system and lambda calculus for both Zd-linear maps and projective Cliffords, and prove that these type systems have a sound denotational semantics in terms of the relevant categories. Finally, we explore what it means to program with projective Cliffords through a number of examples and programming constructions. Copyright © 2024, The Authors. All rights reserved.

关键词： Semantics

Condensed Encodings of Projective Clifford Operations in Arbitrary Dimension

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arXiv 2024年

作者： Winnick, Sam Paykin, Jennifer Institute for Quantum Computing Department of Combinatorics & Optimization University of Waterloo Canada Intel Labs Intel Corporation United States

We provide a careful analysis of the structure theorem for the n-qudit projective Clifford group and various encoding schemes for its elements. In particular, we derive formulas for evaluation, composition, and inversion. Our results apply to all integers d ≥ 2, most notably the even case. Copyright © 2024, The Authors. All rights reserved.

关键词： Encoding (symbols)

quantum State Preparation with Optimal T-Count

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arXiv 2024年

作者： Gosset, David Kothari, Robin Wu, Kewen Google Quantum AI United States Department of Combinatorics and Optimization Institute for Quantum Computing University of Waterloo Canada Perimeter Institute for Theoretical Physics Canada University of California Berkeley United States

How many T gates are needed to approximate an arbitrary n-qubit quantum state to within error Ε? Improving prior work of Low, Kliuchnikov, and Schaeffer, we show that the optimal asymptotic scaling is Θ (Formula Presented) if we allow ancilla qubits. We also show that this is the optimal T -count for implementing an arbitrary diagonal n-qubit unitary to within error Ε. We describe applications in which a tensor product of many single-qubit unitaries can be synthesized in parallel for the price of one. Copyright © 2024, The Authors. All rights reserved.

关键词： Qubits

On the Space of Coefficients of a Feedforward Neural Network

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On the Space of Coefficients of a Feedforward Neural Network

International Joint Conference on Neural Networks (IJCNN)

作者： Dinesh Valluri Rory Campbell Department of Combinatorics and Optimization Institute for Quantum Computing University of Waterloo Waterloo Canada Department of Computer Science Western University London Canada

We establish the conditions for ‘equivalent neural networks’ - neural networks with different weights, biases, and threshold functions which result in the same associated function. We prove that given a neural network $\mathcal{N}$ with piecewise linear activation, the space of coefficients describing all equivalent neural networks is given by a semialgebraic set. This result is obtained by studying different representations of a given piecewise linear function using the Tarski-Seidenberg theorem. Given a neural architecture and an initial neural network $\mathcal{N}_{0}$ , specified by coefficients, we give an algorithm to compute inequalities defining the corresponding semiaglebraic sets. These algorithms are based on the rules of max-plus algebra.

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Triply Efficient Shadow Tomography

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PRX quantum 2025年第1期6卷 010336-010336页

作者： Robbie King David Gosset Robin Kothari Ryan Babbush Google Quantum AI Venice California USA California Institute of Technology Pasadena California USA IQC and Department of Combinatorics and Optimization University of Waterloo Canada Perimeter Institute for Theoretical Physics Waterloo Canada

We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include k-body fermionic or k-local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for k-local Pauli operators with constant k. However, we prove that for k-body fermionic observables and the full set of n-qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.

关键词： Tomography

AN OPERATOR-ALGEBRAIC FORMULATION OF SELF-TESTING

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arXiv 2023年

作者： Paddock, Connor Slofstra, William Zhao, Yuming Zhou, Yangchen Institute for Quantum Computing University of Waterloo Canada Department of Combinatorics & Optimization University of Waterloo Canada Department of Pure Mathematics University of Waterloo Canada

We give a new definition of self-testing for correlations in terms of states on C∗-algebras. We show that this definition is equivalent to the standard definition for any class of finite-dimensional quantum models which is closed, provided that the correlation is extremal and has a full-rank model in the class. This last condition automatically holds for the class of POVM quantum models, but does not necessarily hold for the class of projective models by a result of Baptista, Chen, Kaniewski, Lolck, Mančinska, Gabelgaard Nielsen, and Schmidt. For extremal binary correlations and for extremal synchronous correlations, we show that any self-test for projective models is a self-test for POVM models. The question of whether there is a self-test for projective models which is not a self-test for POVM models remains open. An advantage of our new definition is that it extends naturally to commuting operator models. We show that an extremal correlation is a self-test for finite-dimensional quantum models if and only if it is a self-test for finite-dimensional commuting operator models, and also observe that many known finite-dimensional self-tests are in fact self-tests for infinite-dimensional commuting operator models. Copyright © 2023, The Authors. All rights reserved.

关键词： Algebra

Rate-Distortion Theory for Mixed States

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Rate-Distortion Theory for Mixed States

2023 IEEE International Symposium on Information Theory, ISIT 2023

作者： Khanian, Zahra Baghali Kuroiwa, Kohdai Leung, Debbie Technical University of Munich Munich Center for Quantum Science and Technology Zentrum Mathematik Garching85748 Germany Institute for Quantum Computing University of Waterloo ONN2L 3G1 Canada University of Waterloo Department of Combinatorics and Optimization Canada Perimeter Institute for Theoretical Physics ONN2L 2Y5 Canada

ISBN: (纸本)9781665475549

In this paper we consider the compression of asymptotically many i.i.d. copies of ensembles of mixed quantum states where the encoder has access to a side information system. This source is equivalently defined as a classical-quantum state, namely, a quantum system correlated with a classical system playing the role of an inaccessible reference system. The figure of merit is evaluated based on per-copy or local error criterion. The rate-distortion theory aims to reveal the trade-off between the compression rate and the per-copy error. The optimal trade-off can be characterized by the rate-distortion function, which is the best rate given a certain distortion. In this paper, we analyze the rate-distortion functions of mixed-state compression. We find the rate-distortion functions in the entanglement-assisted and unassisted scenarios, in terms of a single-letter mutual information quantity and the regularized entanglement of purification, respectively. © 2023 IEEE.

关键词： Signal distortion

On the Power of quantum Distributed Proofs

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arXiv 2024年

作者： Hasegawa, Atsuya Kundu, Srijita Nishimura, Harumichi Graduate School of Information Science and Technology The University of Tokyo Japan Institute for Quantum Computing Department of Combinatorics and Optimization University of Waterloo Canada Graduate School of Informatics Nagoya University Japan

quantum nondeterministic distributed computing was recently introduced as dQMA (distributed quantum Merlin-Arthur) protocols by Fraigniaud, Le Gall, Nishimura and Paz (ITCS 2021). In dQMA protocols, with the help of quantum proofs and local communication, nodes on a network verify a global property of the network. Fraigniaud et al. showed that, when the network size is small, there exists an exponential separation in proof size between distributed classical and quantum verification protocols, for the equality problem, where the verifiers check if all the data owned by a subset of them are identical. In this paper, we further investigate and characterize the power of the dQMA protocols for various decision problems. First, we give a more efficient dQMA protocol for the equality problem with a simpler analysis. This is done by adding a symmetrization step on each node and exploiting properties of the permutation test, which is a generalization of the SWAP test. We also show a quantum advantage for the equality problem on path networks still persists even when the network size is large, by considering "relay points" between extreme nodes. Second, we show that even in a general network, there exist efficient dQMA protocols for the ranking verification problem, the Hamming distance problem, and more problems that derive from efficient quantum one-way communication protocols. Third, in a line network, we construct an efficient dQMA protocol for a problem that has an efficient two-party QMA communication protocol. Finally, we obtain the first lower bounds on the proof and communication cost of dQMA protocols. To prove a lower bound on the equality problem, we show any dQMA protocol with an entangled proof between nodes can be simulated with a dQMA protocol with a separable proof between nodes by using a QMA communication-complete problem introduced by Raz and Shpilka (CCC 2004). Copyright © 2024, The Authors. All rights reserved.

关键词： Hamming distance

quantum advantage from measurement-induced entanglement in random shallow circuits

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arXiv 2024年

作者： Watts, Adam Bene Gosset, David Liu, Yinchen Soleimanifar, Mehdi Department of Pure Mathematics University of Waterloo Canada Institute for Quantum Computing University of Waterloo Canada Department of Combinatorics and Optimization University of Waterloo Canada Perimeter Institute for Theoretical Physics Waterloo Canada California Institute of Technology United States

We study random constant-depth quantum circuits in a two-dimensional architecture. While these circuits only produce entanglement between nearby qubits on the lattice, long-range entanglement can be generated by measuring a subset of the qubits of the output state. It is conjectured that this long-range measurement-induced entanglement (MIE) proliferates when the circuit depth is at least a constant critical value d∗. For circuits composed of Haar-random two-qubit gates, it is also believed that this coincides with a quantum advantage phase transition in the classical hardness of sampling from the output distribution. Here we provide evidence for a quantum advantage phase transition in the setting of random Clifford circuits. Our work extends the scope of recent separations between the computational power of constant-depth quantum and classical circuits, demonstrating that this kind of advantage is present in canonical random circuit sampling tasks. In particular, we show that in any architecture of random shallow Clifford circuits, the presence of long-range MIE gives rise to an unconditional quantum advantage. In contrast, any depth-d 2D quantum circuit that satisfies a short-range MIE property can be classically simulated efficiently and with depth O(d). Finally, we introduce a two-dimensional, depth-2, "coarse-grained" circuit architecture, composed of random Clifford gates acting on O(log(n)) qubits, for which we prove the existence of long-range MIE and establish an unconditional quantum advantage. © 2024, CC BY.

关键词： Qubits