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Generic Nonadditivity of quantum Capacity in Simple Channels

Physical Review Letters 2023年第20期130卷 200801-200801页

作者： Felix Leditzky Debbie Leung Vikesh Siddhu Graeme Smith John A. Smolin Department of Mathematics and IQUIST University of Illinois at Urbana-Champaign Urbana Illinois 61801 USA Institute for Quantum Computing and Department of Combinatorics & Optimization University of Waterloo Waterloo Ontario N2L 3G1 Canada and Perimeter Institute for Theoretical Physics Waterloo Ontario N2L 2Y5 Canada. Institute for Quantum Computing and Department of Combinatorics & Optimization University of Waterloo Waterloo Ontario N2L 3G1 Canada and Perimeter Institute for Theoretical Physics Waterloo Ontario N2L 2Y5 Canada. JILA University of Colorado/NIST 440 UCB Boulder Colorado 80309 USA and Department of Physics and Quantum Computing Group Carnegie Mellon University Pittsburgh Pennsylvania 15213 USA. JILA University of Colorado/NIST 440 UCB Boulder Colorado 80309 USA and Department of Physics and Center for Theory of Quantum Matter University of Colorado Boulder Colorado 80309 USA. IBM Quantum IBM T.J. Watson Research Center Yorktown Heights New York 10598 USA.

Determining capacities of quantum channels is a fundamental question in quantum information theory. Despite having rigorous coding theorems quantifying the flow of information across quantum channels, their capacities are poorly understood due to superadditivity effects. Studying these phenomena is important for deepening our understanding of quantum information, yet simple and clean examples of superadditive channels are scarce. Here we study a family of channels called platypus channels. Its simplest member, a qutrit channel, is shown to display superadditivity of coherent information when used jointly with a variety of qubit channels. Higher-dimensional family members display superadditivity of quantum capacity together with an erasure channel. Subject to the “spin-alignment conjecture” introduced in our companion paper [F. Leditzky, D. Leung, V. Siddhu, G. Smith, and J. A. Smolin, The platypus of the quantum channel zoo, IEEE Transactions on Information Theory (IEEE, 2023), 10.1109/TIT.2023.3245985], our results on superadditivity of quantum capacity extend to lower-dimensional channels as well as larger parameter ranges. In particular, superadditivity occurs between two weakly additive channels each with large capacity on their own, in stark contrast to previous results. Remarkably, a single, novel transmission strategy achieves superadditivity in all examples. Our results show that superadditivity is much more prevalent than previously thought. It can occur across a wide variety of channels, even when both participating channels have large quantum capacity.

关键词： quantum channels quantum communication

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Lower bounds for learning quantum states with single-copy measurements

arXiv

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arXiv 2022年

作者： Lowe, Angus Nayak, Ashwin Department of Combinatorics & Optimization Institute for Quantum Computing University of Waterloo 200 University Ave. W. WaterlooONN2L 3G1 Canada

We study the problems of quantum tomography and shadow tomography using measurements performed on individual, identical copies of an unknown d-dimensional state. We first revisit known lower bounds [HHJ+17] on quantum tomography with accuracy Ε in trace distance, when the measurement choices are independent of previously observed outcomes, i.e., they are nonadaptive. We give a succinct proof of these results through the χ2-divergence between suitable distributions. Unlike prior work, we do not require that the measurements be given by rank-one operators. This leads to stronger lower bounds when the learner uses measurements with a constant number of outcomes (e.g., two-outcome measurements). In particular, this rigorously establishes the optimality of the folklore "Pauli tomography" algorithm in terms of its sample complexity. We also derive novel bounds of Ω(r2d/Ε2) and Ω(r2d2/Ε2) for learning rank r states using arbitrary and constant-outcome measurements, respectively, in the nonadaptive case. In addition to the sample complexity, a resource of practical significance for learning quantum states is the number of unique measurement settings required (i.e., the number of different measurements used by an algorithm, each possibly with an arbitrary number of outcomes). Motivated by this consideration, we employ concentration of measure of χ2-divergence of suitable distributions to extend our lower bounds to the case where the learner performs possibly adaptive measurements from a fixed set of exp(O(d)) possible measurements. This implies in particular that adaptivity does not give us any advantage using single-copy measurements that are efficiently implementable. We also obtain a similar bound in the case where the goal is to predict the expectation values of a given sequence of observables, a task known as shadow tomography. Finally, in the case of adaptive, single-copy measurements implementable with polynomial-size circuits, we prove that a straightforward strategy

关键词： Tomography

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Synthesis and Arithmetic of Single Qutrit Circuits

arXiv

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arXiv 2023年

作者： Kalra, Amolak Ratan Mosca, Michele Valluri, Dinesh Institute for Quantum Computing University of Waterloo WaterlooON Canada David R. Cheriton School of Computer Science University of Waterloo WaterlooON Canada Perimeter Institute for Theoretical Physics WaterlooON Canada Dept. of Combinatorics & Optimization University of Waterloo WaterlooON Canada

In this paper we study single qutrit circuits consisting of words over the Clifford+D cyclotomic gate set, where D = diag(±ξa, ±ξb, ±ξc), ξ is a primitive 9-th root of unity and a, b, c are integers. We characterize classes of qutrit unit vectors z with entries in [ξ, χ1 ] based on the possibility of reducing their smallest denominator exponent (sde) with respect to χ := 1 − ξ, by acting an appropriate gate in Clifford+D. We do this by studying the notion of ‘derivatives mod 3’ of an arbitrary element of [ξ] and using it to study the smallest denominator exponent of HDz where H is the qutrit Hadamard gate and D ∈ D. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford+D gates naturally arise as gates with sde 0 and 3 in the group U(3, [ξ, χ1 ]) of 3 × 3 unitaries with entries in [ξ, χ1 ]. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford+R and recover the previous exact synthesis algorithm in [14]. The framework developed to formulate qutrit gate synthesis for Clifford+D extends to qudits of arbitrary prime power. © 2023, CC BY.

关键词： Digital arithmetic

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Generalized quantum Stein's lemma: Redeeming second law of resource theories

arXiv

引用

arXiv 2024年

作者： Yamasaki, Hayata Kuroiwa, Kohdai Department of Physics Graduate School of Science The Univerisity of Tokyo 7-3-1 Hongo Bunkyo-ku Tokyo113-0033 Japan Institute for Quantum Computing Department of Combinatorics and Optimization University of Waterloo ONN2L 3G1 Canada Perimeter Institute for Theoretical Physics ONN2L 2Y5 Canada

The second law lies at the heart of thermodynamics, characterizing the convertibility of thermodynamic states by a single quantity, the entropy. A fundamental question in quantum information theory is whether one can formulate an analogous second law characterizing the convertibility of resources for quantum information processing. In 2008, a promising formulation was proposed, where quantum-resource convertibility is characterized by the optimal performance of a variant of another fundamental task in quantum information processing, quantum hypothesis testing. The core of this formulation was to prove a lemma that identifies a quantity indicating the optimal performance of this task-the generalized quantum Stein's lemma-to seek out a counterpart of the thermodynamic entropy in quantum information processing. However, in 2023, a logical gap was found in the existing proof of the generalized quantum Stein's lemma, throwing into question once again whether such a formulation is possible at all. In this work, we construct a proof of the generalized quantum Stein's lemma by developing alternative techniques to circumvent the logical gap of the existing analysis. With our proof, we redeem the formulation of quantum resource theories equipped with the second law as desired. These results affirmatively settle the fundamental question about the possibility of bridging the analogy between thermodynamics and quantum information theory. Copyright © 2024, The Authors. All rights reserved.

关键词： Entropy

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A rapidly mixing Markov chain from any gapped quantum many-body system

arXiv

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arXiv 2022年

作者： Bravyi, Sergey Carleo, Giuseppe Gosset, David Liu, Yinchen IBM Quantum IBM T.J. Watson Research Center Yorktown Heights United States Institute of Physics LausanneCH-1015 Switzerland Department of Combinatorics and Optimization Institute for Quantum Computing University of Waterloo Canada Perimeter Institute for Theoretical Physics Waterloo Canada

We consider the computational task of sampling a bit string x from a distribution π(x) = |〈x|ψ〉|2, where ψ is the unique ground state of a local Hamiltonian H. Our main result describes a direct link between the inverse spectral gap of H and the mixing time of an associated continuous-time Markov Chain with steady state π. The Markov Chain can be implemented efficiently whenever ratios of ground state amplitudes 〈y|ψ〉/〈x|ψ〉 are efficiently computable, the spectral gap of H is at least inverse polynomial in the system size, and the starting state of the chain satisfies a mild technical condition that can be efficiently checked. This extends a previously known relationship between sign-problem free Hamiltonians and Markov chains. The tool which enables this generalization is the so-called fixed-node Hamiltonian construction, previously used in quantum Monte Carlo simulations to address the fermionic sign problem. We implement the proposed sampling algorithm numerically and use it to sample from the ground state of Haldane-Shastry Hamiltonian with up to 56 qubits. We observe empirically that our Markov chain based on the fixed-node Hamiltonian mixes more rapidly than the standard Metropolis-Hastings Markov chain. © 2022, CC BY.

关键词： Hamiltonians

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Synthesizing efficient circuits for Hamiltonian simulation

arXiv

引用

arXiv 2022年

作者： Mukhopadhyay, Priyanka Wiebe, Nathan Zhang, Hong Tao Institute for Quantum Computing University of Waterloo Canada Department of Combinatorics and Optimization University of Waterloo Canada Department of Computer Science University of Toronto Canada Pacific Northwest National Laboratory United States Department of Mathematics University of Toronto Canada

We provide a new approach for compiling quantum simulation circuits that appear in Trotter, qDRIFT and multi-product formulas to Clifford and non-Clifford operations that can reduce the number of non-Clifford operations by a factor of up to 4. In fact, the total number of gates reduce in many cases. We show that it is possible to implement an exponentiated sum of commuting Paulis with at most m (controlled)-rotation gates, where m is the number of distinct non-zero eigenvalues (ignoring sign). Thus we can collect mutually commuting Hamiltonian terms into groups that satisfy one of several symmetries identified in this work which allow an inexpensive simulation of the entire group of terms. We further show that the cost can in some cases be reduced by partially allocating Hamiltonian terms to several groups and provide a polynomial time classical algorithm that can greedily allocate the terms to appropriate groupings. We further specifically discuss these optimizations for the case of fermionic dynamics and provide extensive numerical simulations for qDRIFT of our grouping strategy to 6 and 4-qubit Heisenberg models, LiH, H2 and observe a factor of 1.8-3.2 reduction in the number of non-Clifford gates. This suggests Trotter-based simulation of chemistry in second quantization may be even more practical than previously believed. © 2022, CC BY.

关键词： Hamiltonians

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Achieving Fault Tolerance on Capped Color Codes with Few Ancillas

引用

PRX quantum 2022年第3期3卷 030322-030322页

作者： Theerapat Tansuwannont Debbie Leung Institute for Quantum Computing and Department of Physics and Astronomy University of Waterloo Waterloo Ontario N2L 3G1 Canada Duke Quantum Center and Department of Electrical and Computer Engineering Duke University Durham North Carolina 27708 USA Institute for Quantum Computing and Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario N2L 3G1 Canada Perimeter Institute for Theoretical Physics Waterloo Ontario N2L 2Y5 Canada

Attaining fault tolerance while maintaining low overhead is one of the main challenges in a practical implementation of quantum circuits. One major technique that can overcome this problem is the flag technique, in which high-weight errors arising from a few faults can be detected by a few ancillas and distinguished using subsequent syndrome measurements. The technique can be further improved using the fact that, for some families of codes, errors of any weight are logically equivalent if they have the same syndrome and weight parity, as shown in our previous work [Tansuwannont and Leung, Phys. Rev. A 104, 042410 (2021)]. In this work, we develop a notion of distinguishable fault set that captures both concepts of flags and weight parities, and extend the use of weight parities in error correction to families of capped and recursive capped color codes. We also develop fault-tolerant protocols for error correction, measurement, state preparation, and logical T-gate implementation via code switching, which are sufficient for performing fault-tolerant Clifford computation on a capped color code, and performing fault-tolerant universal quantum computation on a recursive capped color code. Our protocols for a capped or a recursive capped color code of any distance require only two ancillas, assuming that the ancillas can be reused. The concept of distinguishable fault set also leads to a generalization of the definitions of fault-tolerant gadgets proposed by Aliferis, Gottesman, and Preskill.

关键词： quantum computation quantum error correction

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ARKHIPOV'S THEOREM, GRAPH MINORS, AND LINEAR SYSTEM NONLOCAL GAMES

arXiv

引用

arXiv 2022年

作者： Paddock, Connor Russo, Vincent Silverthorne, Turner Slofstra, William Institute for Quantum Computing University of Waterloo Canada Department of Combinatorics & Optimization University of Waterloo Canada Department of Pure Mathematics University of Waterloo Canada Department of Mathematics University of Toronto Canada Modellicity Inc. Canada Unitary Fund Canada

The perfect quantum strategies of a linear system game correspond to certain representations of its solution group. We study the solution groups of graph incidence games, which are linear system games in which the underlying linear system is the incidence system of a (non-properly) two-coloured graph. While it is undecidable to determine whether a general linear system game has a perfect quantum strategy, for graph incidence games this problem is solved by Arkhipov's theorem, which states that the graph incidence game of a connected graph has a perfect quantum strategy if and only if it either has a perfect classical strategy, or the graph is nonplanar. Arkhipov's criterion can be rephrased as a forbidden minor condition on connected two-coloured graphs. We extend Arkhipov's theorem by showing that, for graph incidence games of connected two-coloured graphs, every quotient closed property of the solution group has a forbidden minor characterization. We rederive Arkhipov's theorem from the group theoretic point of view, and then find the forbidden minors for two new properties: finiteness and abelianness. Our methods are entirely combinatorial, and finding the forbidden minors for other quotient closed properties seems to be an interesting combinatorial problem. Copyright © 2022, The Authors. All rights reserved.

关键词： Linear systems

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Beyond product state approximations for a quantum analogue of max cut 15

Beyond product state approximations for a quantum analogue o...

引用

15th Conference on the Theory of quantum Computation, Communication and Cryptography, TQC 2020

作者： Anshu, Anurag Gosset, David Morenz, Karen Institute for Quantum Computing University of Waterloo Canada Department of Combinatorics and Optimization University of Waterloo Canada Perimeter Institute for Theoretical Physics Waterloo Canada Department of Chemistry University of Toronto Canada

ISBN: (纸本)9783959771467

We consider a computational problem where the goal is to approximate the maximum eigenvalue of a two-local Hamiltonian that describes Heisenberg interactions between qubits located at the vertices of a graph. Previous work has shed light on this problems approximability by product states. For any instance of this problem the maximum energy attained by a product state is lower bounded by the Max Cut of the graph and upper bounded by the standard Goemans-Williamson semidefinite programming relaxation of it. Gharibian and Parekh described an efficient classical approximation algorithm for this problem which outputs a product state with energy at least 0.498 times the maximum eigenvalue in the worst case, and observe that there exist instances where the best product state has energy 1/2 of optimal. We investigate approximation algorithms with performance exceeding this limitation which are based on optimizing over tensor products of few-qubit states and shallow quantum circuits. We provide an efficient classical algorithm which achieves an approximation ratio of at least 0.53 in the worst case. We also show that for any instance defined by a 3 or 4-regular graph, there is an efficiently computable shallow quantum circuit that prepares a state with energy larger than the best product state (larger even than its semidefinite programming relaxation). © Anurag Anshu, David Gosset, and Karen Morenz.

关键词： Approximation algorithms

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Reducing the CNOT count for Clifford+T circuits on NISQ architectures

TechRxiv

引用

TechRxiv 2022年

作者： Mukhopadhyay, Priyanka Gheorghiu, Vlad Huang, Jiaxin Li, Sarah Meng Mosca, Michele Institute for Quantum Computing University of Waterloo WaterlooON Canada Department of Combinatorics and Optimization University of Waterloo WaterlooON Canada Perimeter Institute for Theoretical Physics WaterlooON Canada SoftwareQ Inc. KitchenerON Canada Faculty of Mathematics University of Waterloo Canada

While mapping a quantum circuit to the physical layer one has to consider the numerous constraints imposed by the underlying hardware architecture. Connectivity of the physical qubits is one such constraint that restricts two-qubit operations, such as CNOT, to "connected" qubits. SWAP gates can be used to place the logical qubits on admissible physical qubits, but they entail a significant increase in CNOT-count. In this paper we consider the problem of reducing the CNOT-count in Clifford+T circuits on connectivity constrained architectures, like noisy intermediate-scale quantum (NISQ) computing devices. We "slice" the circuit at the position of Hadamard gates and "build" the intermediate {CNOT, T} sub-circuits using Steiner trees, significantly improving on previous methods. We compared the performance of our algorithms while mapping different benchmark and random circuits to some well-known architectures such as 9-qubit square grid, 16-qubit square grid, Rigetti 16-qubit Aspen, 16-qubit IBM QX5 and 20-qubit IBM Tokyo. Our methods give less CNOT-count compared to Qiskit transpiler as well as using SWAP gates. Assuming most of the errors in a NISQ circuit implementation are due to CNOT errors, then our method would allow circuits with few times more CNOT gates be reliably implemented than the previous methods would permit. © 2022, CC BY-NC-SA.

关键词： Qubits

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