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arXiv 2022年

作者： Leditzky, Felix Leung, Debbie Siddhu, Vikesh Smith, Graeme Smolin, John A. Department of Mathematics IQUIST University of Illinois at Urbana-Champaign United States Institute for Quantum Computing Department of Combinatorics and Optimization University of Waterloo Canada Perimeter Institute for Theoretical Physics Canada JILA University of Colorado Boulder United States Department of Physics & Quantum Computing Group Carnegie Mellon University United States CTQM Department of Physics University of Colorado Boulder United States IBM Quantum IBM T.J. Watson Research Center United States

Understanding quantum channels and the strange behavior of their capacities is a key objective of quantum information theory. Here we study a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. As the simplest example from this family, we focus on a qutrit-to-qutrit channel that is intuitively obtained by hybridizing together a simple degradable channel and a completely useless qubit channel. Such hybridizing makes this channel’s capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel does not belong to any class for which the underlying information quantities are known to be additive. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This "spin alignment conjecture," which may be of independent interest, is proved in certain special cases and additional numerical evidence for its validity is provided. Finally, we generalize the qutrit channel in two ways, and the resulting channels and their capacities display similarly rich behavior. In the companion paper [34], we further show that the qutrit channel demonstrates superadditivity when transmitting quantum information jointly with a variety of assisting channels, in a manner unknown before. Copyright © 2022, The Authors. All rights reserved.

关键词： quantum optics

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Generic Nonadditivity of quantum Capacity in Simple Channels

引用

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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Flag fault-tolerant error correction, measurement, and quantum computation for cyclic CSS codes

arXiv

引用

arXiv 2018年

作者： Tansuwannont, Theerapat Chamberland, Christopher Leung, Debbie Institute for Quantum Computing Department of Physics and Astronomy University of Waterloo WaterlooONN2L 3G1 Canada IBM T. J. Watson Research Center Yorktown HeightsNY10598 United States Institute for Quantum Computing Department of Combinatorics and Optimization University of Waterloo WaterlooONN2L 3G1 Canada

Flag qubits have recently been proposed in syndrome extraction circuits to detect high-weight errors arising from fewer faults. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to-date. In this work, we prove some critical properties of CSS codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme. We then develop fault-tolerant protocols as well as a family of circuits for flag fault-tolerant error correction and operator measurement, requiring only four ancilla qubits and applicable to cyclic CSS codes of distance 3. The measurement protocol can be further used for logical Clifford gate implementation via quantum gate teleportation. We also provide examples of cyclic CSS codes with large encoding rates. Copyright © 2018, The Authors. All rights reserved.

关键词： Error correction

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Proper versus Improper quantum PAC Learning

arXiv

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arXiv 2024年

作者： Nayak, Ashwin Sinha, Pulkit Department of Combinatorics and Optimization Institute for Quantum Computing University of Waterloo 200 University Ave. W. WaterlooONN2L 3G1 Canada School of Computer Science Institute for Quantum Computing University of Waterloo 200 University Ave. W. WaterlooONN2L 3G1 Canada

A basic question in the PAC model of learning is whether proper learning is harder than improper learning. In the classical case, the answer to this question, with respect to sample complexity, is known to depend on the concept class. While there are concept classes for which the two modes of learning have the same complexity, there are examples of concept classes with VC dimension d that have sample complexity Ω ( dǫ log 1ǫ ) for proper learning with error ǫ, while the complexity for improper learning is O( dǫ ). One such example arises from the Coupon Collector problem. Motivated by the efficiency of proper versus improper learning with quantum samples, Arunachalam, Belovs, Childs, Kothari, Rosmanis, and de Wolf [2] studied an analogue, the quantum Coupon Collector problem. Curiously, they discovered that for learning size k subsets of [n] the problem has sample complexity Θ(k log min {k, n − k + 1}), in contrast with the complexity of Θ(k log k) for Coupon Collector. This effectively negates the possibility of a separation between the two modes of learning via the quantum problem, and Arunachalam et al. posed the possibility of such a separation as an open question. In this work, we first present an algorithm for the quantum Coupon Collector problem with sample complexity that matches the sharper lower bound of (1 − ok(1))k ln min {k, n − k + 1} shown recently by Bab Hadiashar, Nayak, and Sinha [8], for the entire range of the parameter k. Next, we devise a variant of the problem, the quantum Padded Coupon Collector. We prove that its sample complexity matches that of the classical Coupon Collector problem for both modes of learning, thereby exhibiting the same asymptotic separation between proper and improper quantum learning as mentioned above. The techniques we develop in the process can be directly applied to any form of padded quantum data. We hope that padding can more generally lift other forms of classical learning behaviour to the quantum setting. © 2024

关键词： Learning systems

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Synthesis and Arithmetic of Single Qutrit Circuits

arXiv

引用

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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Communication complexity of one-shot remote state preparation

arXiv

引用

arXiv 2018年

作者： Hadiashar, Shima Bab Nayak, Ashwin Renner, Renato Department of Combinatorics and Optimization Institute for Quantum Computing University of Waterloo 200 University Ave. W WaterlooONN2L 3G1 Canada Institute for Theoretical Physics Eth Zurich Wolfgang-Pauli-Str. 27 Zurich8093

quantum teleportation uses prior shared entanglement and classical communication to send an unknown quantum state from one party to another. Remote state preparation (RSP) is a similar distributed task in which the sender knows the entire classical description of the state to be sent. (This may also be viewed as the task of non-oblivious compression of a single sample from an ensemble of quantum states.) We study the communication complexity of approximate remote state preparation, in which the goal is to prepare an approximation of the desired quantum state. Jain [Quant. Inf. & Comp., 2006] showed that the worst-case communication complexity of approximate RSP can be bounded from above in terms of the maximum possible information in an encoding. He also showed that this quantity is a lower bound for communication complexity of (exact) remote state preparation. In this work, we tightly characterize the worst-case and average-case communication complexity of remote state preparation in terms of non-Asymptotic information-Theoretic quantities. We also show that the average-case communication complexity of RSP can be much smaller than the worst-case one. In the process, we show that n bits cannot be communicated with less than n transmitted bits in LOCC protocols. This strengthens a result due to Nayak and Salzman [J. ACM, 2006] and may be of independent interest. Copyright © 2018, The Authors. All rights reserved.

关键词： Computational complexity

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Generating graph states with a single quantum emitter and the minimum number of fusions

arXiv

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arXiv 2024年

作者： Löbl, Matthias C. Pettersson, Love A. Jena, Andrew Dellantonio, Luca Paesani, Stefano Sørensen, Anders S. The Niels Bohr Institute University of Copenhagen Blegdamsvej 17 Copenhagen ØDK-2100 Denmark Department of Combinatorics and Optimization University of Waterloo Institute for Quantum Computing University of Waterloo Canada Department of Physics and Astronomy University of Exeter Stocker Road ExeterEX4 4QL United Kingdom NNF Quantum Computing Programme Niels Bohr Institute University of Copenhagen Blegdamsvej 17 Copenhagen ØDK-2100 Denmark

Graph states are the key resources for measurement- and fusion-based quantum computing with photons, yet their creation is experimentally challenging. We optimize a hybrid graph-state generation scheme using a single quantum emitter and linear optics Bell-state measurements, called fusions. We first generate a restricted class of states from a single quantum emitter and then apply fusions to create a target graph state, where we use a dynamic programming approach to find the construction that requires the lowest possible number of fusions. Our analysis yields a lookup table for constructing ∼ 2.8×107 non-isomorphic graph states with the minimum number of fusions. The lookup table covers all graph states with up to eight qubits and several other ones with up to 14 qubits. We present construction protocols of selected graph states and provide the lookup table. For large graph states that are not in the lookup table, we derive bounds for the required number of fusions using graph-theoretic properties. Finally, we use the lookup table to search for the best graph codes for loss-tolerant encodings, given a fixed number of fusions for their construction. Copyright © 2024, The Authors. All rights reserved.

关键词： Qubits

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Sample-optimal classical shadows for pure states

arXiv

引用

arXiv 2022年

作者： Grier, Daniel Pashayan, Hakop Schaeffer, Luke Department of Mathematics Department of Computer Science and Engineering UC San Diego United States Institute for Quantum Computing University of Waterloo Canada Department of Combinatorics and Optimization University of Waterloo Canada Perimeter Institute for Theoretical Physics Waterloo Canada Dahlem Center for Complex Quantum Systems Freie Universität Berlin Germany Joint Center for Quantum Information and Computer Science University of Maryland College Park United States

We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state ρ in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate Tr(Oρ) for any Hermitian observable O to within additive error ϵ provided Tr(O2) ≤ B and O = 1._ Our main result applies to the joint measurement setting, where we show Θ˜(√Bϵ−1 + ϵ−2) samples of ρ are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of ρ can be compressed for observable estimation. In the independent measurement setting, we show that O(√Bdϵ−1 + ϵ−2) samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator. © 2022, CC BY.

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How to Simulate quantum Measurement without computing Marginals

引用

Physical Review Letters 2022年第22期128卷 220503-220503页

作者： Sergey Bravyi David Gosset Yinchen Liu IBM Quantum IBM T.J. Watson Research Center Yorktown Heights New York 10598 USA Department of Combinatorics and Optimization University of Waterloo Waterloo ON N2L 3G1 Canada Institute for Quantum Computing University of Waterloo Waterloo ON N2L 3G1 Canada Perimeter Institute for Theoretical Physics Waterloo ON N2L 2Y5 Canada

We describe and analyze algorithms for classically simulating measurement of an n-qubit quantum state in the standard basis, that is, sampling a bit string from the probability distribution determined by the Born rule. Our algorithms reduce the sampling task to computing poly(n) amplitudes of n-qubit states; unlike previously known techniques they do not require computation of marginal probabilities. Two classes of quantum states are considered: output states of polynomial-size quantum circuits, and ground states of local Hamiltonians with an inverse polynomial spectral gap. We show that our algorithms can significantly accelerate quantum circuit simulations based on tensor network contraction or low-rank stabilizer decompositions. As another striking consequence we obtain the first efficient classical simulation algorithm for measurement-based quantum computation with the surface code resource state on any planar graph and any schedule of measurements.

关键词： Measurement-based quantum computing quantum algorithms quantum benchmarking quantum simulation

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Minimal qubit representations of Hamiltonians via conserved charges

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Physical Review A 2024年第2期109卷 022618-022618页

作者： Lane G. Gunderman Andrew Jena Luca Dellantonio Institute for Quantum Computing University of Waterloo Waterloo Ontario Canada N2L 3G1 Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario Canada N2L 3G1 Department of Physics and Astronomy University of Exeter Stocker Road Exeter EX4 4QL United Kingdom

In the last years, we have been witnessing a tremendous push to demonstrate that quantum computers can solve classically intractable problems. This effort, initially focused on the hardware, progressively included the simplification of the models to be simulated. We consider Hamiltonians written in terms of Pauli operators and systematically cut all qubits that are not essential to simulate the system. Our approach is universally applicable and lowers the complexity by first ensuring that the largest possible portion of the Hilbert space becomes irrelevant, and then by finding and exploiting all conserved charges of the system, i.e., symmetries that can be expressed as Pauli operators. Remarkably, both processes are classically efficient and optimal. To showcase our algorithm, we simplify chemical molecules, lattice gauge theories, and the Hubbard and Kitaev models.

关键词： quantum algorithms & computation quantum computation quantum error correction quantum information theory quantum simulation

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