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Taming Quantum Time Complexity 驯服量子时间复杂性
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-23 DOI: 10.22331/q-2024-08-23-1444
Aleksandrs Belovs, Stacey Jeffery, Duyal Yolcu
Quantum query complexity has several nice properties with respect to composition. First, bounded-error quantum query algorithms can be composed without incurring log factors through error reduction $exactness$. Second, through careful accounting $thriftiness$, the total query complexity is smaller if subroutines are mostly run on cheaper inputs -- a property that is much less obvious in quantum algorithms than in their classical counterparts. While these properties were previously seen through the model of span programs (alternatively, the dual adversary bound), a recent work by two of the authors (Belovs, Yolcu 2023) showed how to achieve these benefits without converting to span programs, by defining $textit{quantum Las Vegas query complexity}$. Independently, recent works, including by one of the authors (Jeffery 2022), have worked towards bringing thriftiness to the more practically significant setting of quantum $time$ complexity.

In this work, we show how to achieve both exactness and thriftiness in the setting of time complexity. We generalize the quantum subroutine composition results of Jeffery 2022 so that, in particular, no error reduction is needed. We give a time complexity version of the well-known result in quantum query complexity, $Q(fcirc g)=mathcal{O}(Q(f)cdot Q(g))$, without log factors.

We achieve this by employing a novel approach to the design of quantum algorithms based on what we call $transducers$, and which we think is of large independent interest. While a span program is a completely different computational model, a transducer is a direct generalisation of a quantum algorithm, which allows for much greater transparency and control. Transducers naturally characterize general state conversion, rather than only decision problems; provide a very simple treatment of other quantum primitives such as quantum walks; and lend themselves well to time complexity analysis.
量子查询复杂性在组成方面有几个很好的特性。首先,有界错误量子查询算法可以通过减少错误的 "精确性"(exactness)进行组合,而不会产生对数因子。其次,通过仔细核算 "漂移性"(thriftiness),如果子程序大多运行在更便宜的输入上,则总查询复杂度会更小--与经典算法相比,量子算法的这一特性并不明显。虽然这些特性以前是通过跨度程序模型(或者说是双重对手约束)看到的,但两位作者最近的一项工作(Belovs, Yolcu 2023)表明了如何通过定义 $textit{quantum Las Vegas query complexity}$,在不转换到跨度程序的情况下实现这些优势。另外,包括作者之一(杰弗里 2022)在内的近期研究致力于将节俭性引入更具实际意义的量子$time$复杂性设置中。在这项工作中,我们展示了如何在时间复杂性中同时实现精确性和节俭性。我们概括了杰弗里 2022 的量子子程序组成结果,因此尤其不需要减少误差。我们给出了量子查询复杂度中著名结果 $Q(fcirc g)=mathcal{O}(Q(f)cdot Q(g))$的时间复杂度版本,不含对数因子。我们采用了一种新颖的方法来设计量子算法,这种方法基于我们称之为 $transducers$的程序,我们认为这种方法具有很大的独立意义。跨度程序是一种完全不同的计算模型,而转换器则是量子算法的直接概括,它允许更大的透明度和控制。量子转换器自然地表征了一般状态转换,而不仅仅是决策问题;为量子行走等其他量子原语提供了非常简单的处理方法;并且非常适合时间复杂性分析。
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引用次数: 0
Classical-to-quantum non-signalling boxes 经典-量子非信号箱
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-22 DOI: 10.22331/q-2024-08-22-1443
Carolina Moreira Ferrera, Robin Simmons, James Purcell, Daniel Collins, Sandu Popescu
Here we introduce the concept of classical input – quantum output (C-Q) non-signalling boxes, a generalisation of the classical input – classical output (C-C) non-signalling boxes. We argue that studying such objects leads to a better understanding of the relation between quantum nonlocality and non-locality beyond quantum mechanics. The main issue discussed in the paper is whether there exist 'genuine' C-Q boxes or all C-Q boxes can be built from objects already known, namely C-C boxes acting on pre-shared entangled quantum particles. We show that large classes of C-Q boxes are non-genuine. In particular, we show that all bi-partite C-Q boxes with outputs that are pure states are non-genuine. We also present various strategies for addressing the general problem, i.e. for multi-partite C-Q boxes which output mixed states, whose answer is still open. Finally, we show that even some very simple non-genuine C-Q boxes require large amounts of C-C nonlocal correlations in order to simulate them.
在这里,我们引入了经典输入-量子输出(C-Q)非信号箱的概念,它是经典输入-经典输出(C-C)非信号箱的概括。我们认为,研究这类对象有助于更好地理解量子非位置性与量子力学之外的非位置性之间的关系。本文讨论的主要问题是,是否存在 "真正的 "C-Q 盒,或者所有的 C-Q 盒都可以从已知的对象(即作用于预先共享的纠缠量子粒子的 C-C 盒)中构建出来。我们证明了很大一类 C-Q 盒是非真正的。特别是,我们证明了所有输出为纯态的二分 C-Q 盒都是非真正的。我们还提出了解决一般问题的各种策略,即对于输出混合状态的多方 C-Q 盒,其答案仍未确定。最后,我们证明,即使是一些非常简单的非真正 C-Q 盒,也需要大量的 C-C 非局部相关性才能模拟它们。
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引用次数: 0
Certification of quantum state functions under partial information 部分信息下的量子态函数认证
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-16 DOI: 10.22331/q-2024-08-16-1442
Leonardo Zambrano, Donato Farina, Egle Pagliaro, Marcio M. Taddei, Antonio Acin
Convex functions of quantum states play a key role in quantum physics, with examples ranging from Bell inequalities to von Neumann entropy. However, in experimental scenarios, direct measurements of these functions are often impractical. We address this issue by introducing two methods for determining rigorous confidence bounds for convex functions based on informationally incomplete measurements. Our approach outperforms existing protocols by providing tighter bounds for a fixed confidence level and number of measurements. We evaluate the performance of our methods using both numerical and experimental data. Our findings demonstrate the efficacy of our approach, paving the way for improved quantum state certification in real-world applications.
量子态的凸函数在量子物理学中起着关键作用,从贝尔不等式到冯-诺依曼熵都是例子。然而,在实验场景中,直接测量这些函数往往是不切实际的。为了解决这个问题,我们介绍了两种基于信息不完全测量确定凸函数严格置信度边界的方法。我们的方法优于现有协议,能为固定置信度和测量次数提供更严格的界限。我们利用数值和实验数据评估了我们方法的性能。我们的研究结果证明了我们方法的有效性,为改进现实世界应用中的量子态认证铺平了道路。
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引用次数: 0
Quantum Reference Frames for Lorentz Symmetry 洛伦兹对称的量子参考框架
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-14 DOI: 10.22331/q-2024-08-14-1440
Luca Apadula, Esteban Castro-Ruiz, Časlav Brukner
Since their first introduction, Quantum Reference Frame (QRF) transformations have been extensively discussed, generalising the covariance of physical laws to the quantum domain. Despite important progress, a formulation of QRF transformations for Lorentz symmetry is still lacking. The present work aims to fill this gap. We first introduce a reformulation of relativistic quantum mechanics independent of any notion of preferred temporal slicing. Based on this, we define transformations that switch between the perspectives of different relativistic QRFs. We introduce a notion of ''quantum Lorentz transformations'' and ''superposition of Lorentz boosts'', acting on the external degrees of freedom of a quantum particle. We analyse two effects, superposition of time dilations and superposition of length contractions, that arise only if the reference frames exhibit both relativistic and quantum-mechanical features. Finally, we discuss how the effects could be observed by measuring the wave-packet extensions from relativistic QRFs.
量子参考框架(QRF)变换自首次提出以来,已被广泛讨论,它将物理定律的协方差推广到量子领域。尽管取得了重要进展,但仍缺乏针对洛伦兹对称性的 QRF 变换的表述。本研究旨在填补这一空白。我们首先介绍了相对论量子力学的重新表述,它独立于任何优先时间切片的概念。在此基础上,我们定义了在不同相对论 QRF 视角之间切换的变换。我们引入了 "量子洛伦兹变换 "和 "洛伦兹叠加 "的概念,它们作用于量子粒子的外部自由度。我们分析了时间膨胀叠加和长度收缩叠加这两种效应,它们只有在参照系同时表现出相对论和量子力学特征时才会出现。最后,我们讨论了如何通过测量相对论 QRF 的波包扩展来观测这些效应。
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引用次数: 0
Study of noise in virtual distillation circuits for quantum error mitigation 研究虚拟蒸馏电路中的噪声以减少量子错误
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-14 DOI: 10.22331/q-2024-08-14-1441
Pontus Vikstål, Giulia Ferrini, Shruti Puri
Virtual distillation has been proposed as an error mitigation protocol for estimating the expectation values of observables in quantum algorithms. It proceeds by creating a cyclic permutation of $M$ noisy copies of a quantum state using a sequence of controlled-swap gates. If the noise does not shift the dominant eigenvector of the density operator away from the ideal state, then the error in expectation-value estimation can be exponentially reduced with $M$. In practice, subsequent error mitigation techniques are required to suppress the effect of noise in the cyclic permutation circuit itself, leading to increased experimental complexity. Here, we perform a careful analysis of the effect of uncorrelated, identical noise in the cyclic permutation circuit and find that the estimation of expectation value of observables are robust against dephasing noise. We support the analytical result with numerical simulations and find that $67%$ of errors are reduced for $M=2$, with physical dephasing error probabilities as high as $10%$. Our results imply that a broad class of quantum algorithms can be implemented with higher accuracy in the near-term with qubit platforms where non-dephasing errors are suppressed, such as superconducting bosonic qubits and Rydberg atoms.
虚拟蒸馏是一种用于估算量子算法中观测值期望值的错误缓解协议。其方法是利用一系列受控交换门创建量子态的 $M$ 噪音副本的循环排列。如果噪声不会使密度算子的主特征向量偏离理想状态,那么期望值估计的误差就会随着 $M$ 的增加而呈指数减少。实际上,后续的误差缓解技术需要抑制循环置换电路本身的噪声影响,从而导致实验复杂度增加。在这里,我们对循环置换电路中不相关的相同噪声的影响进行了仔细分析,发现观测值的期望值估计对去相噪声具有鲁棒性。我们用数值模拟来支持这一分析结果,发现在 $M=2$ 时,误差减少了 $67%$,而物理消隐误差概率高达 $10%$。我们的结果表明,在短期内可以通过抑制非去相干误差的量子比特平台(如超导玻色子量子比特和雷德堡原子),以更高的精度实现各种量子算法。
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引用次数: 0
Approximate Quantum Codes From Long Wormholes 来自长虫洞的近似量子密码
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-14 DOI: 10.22331/q-2024-08-14-1439
Gregory Bentsen, Phuc Nguyen, Brian Swingle
We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the SYK model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, $N$, of fermions goes to infinity. For SYK, the distance scales as $N^{1/2}$, and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. $N^{.99}$, while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property which we dub the no low-energy adiabatically accessible states property and show that these models do have low-energy states that can be prepared adiabatically in a time that does not scale with system size $N$. We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity.
我们讨论了近似量子纠错码系列,它们是由非交换项组成的某些量子多体哈密顿的近退化基态。对于精确编码,纠错条件可以用低温热场双态中双面互信息的消失来表述。我们考虑了近似码的距离概念,要求这种互信息很小,并对 SYK 模型和一系列低阶 SYK 模型的互信息进行了评估。我们对 SYK 模型和一系列低阶 SYK 模型的互信息进行了评估。经过对接近零温度的推断,我们发现当费米子数量 $N$ 变为无穷大时,这两种模型都能产生速率恒定的费米子密码。对 SYK 来说,距离按 $N^{1/2}$ 缩放,而对低阶 SYK 来说,距离可以任意接近线性缩放,例如 $N^{.99}$,同时保持恒定的速率。我们还考虑了无低能琐态性质的类似性质,并将其称为 "无低能绝热可及态性质",结果表明这些模型确实存在可以在不随系统大小 $N$ 变化的时间内绝热制备的低能态。我们讨论了这些密码的全息模型,其中大密码距离是一个简单量子引力模型中出现长虫洞几何的结果。
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引用次数: 0
Measurement-efficient quantum Krylov subspace diagonalisation 测量效率高的量子克雷洛夫子空间对角化
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-13 DOI: 10.22331/q-2024-08-13-1438
Zongkang Zhang, Anbang Wang, Xiaosi Xu, Ying Li
The Krylov subspace methods, being one category of the most important classical numerical methods for linear algebra problems, can be much more powerful when generalised to quantum computing. However, quantum Krylov subspace algorithms are prone to errors due to inevitable statistical fluctuations in quantum measurements. To address this problem, we develop a general theoretical framework to analyse the statistical error and measurement cost. Based on the framework, we propose a quantum algorithm to construct the Hamiltonian-power Krylov subspace that can minimise the measurement cost. In our algorithm, the product of power and Gaussian functions of the Hamiltonian is expressed as an integral of the real-time evolution, such that it can be evaluated on a quantum computer. We compare our algorithm with other established quantum Krylov subspace algorithms in solving two prominent examples. To achieve an error comparable to that of the classical Lanczos algorithm at the same subspace dimension, our algorithm typically requires orders of magnitude fewer measurements than others. Such an improvement can be attributed to the reduced cost of composing projectors onto the ground state. These results show that our algorithm is exceptionally robust to statistical fluctuations and promising for practical applications.
克雷洛夫子空间方法是线性代数问题中最重要的一类经典数值方法,如果推广到量子计算中,其功能会更加强大。然而,由于量子测量中不可避免的统计波动,量子克雷洛夫子空间算法容易出现误差。为了解决这个问题,我们开发了一个分析统计误差和测量成本的通用理论框架。在此框架基础上,我们提出了一种量子算法,用于构建能使测量成本最小化的哈密顿-功率克雷洛夫子空间。在我们的算法中,哈密顿的幂函数和高斯函数的乘积表示为实时演化的积分,因此可以在量子计算机上进行评估。在解决两个突出的例子时,我们将我们的算法与其他成熟的量子克雷洛夫子空间算法进行了比较。在相同的子空间维度下,要达到与经典 Lanczos 算法相当的误差,我们的算法通常需要比其他算法少几个数量级的测量。这种改进可归因于将投影器合成到基态的成本降低了。这些结果表明,我们的算法对统计波动异常稳健,在实际应用中大有可为。
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引用次数: 0
Maximum expectation of observables with restricted purity states 受限纯度态观测值的最大期望值
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-13 DOI: 10.22331/q-2024-08-13-1437
Vikesh Siddhu, John Aaron Smolin
Assessment of practical quantum information processing (QIP) remains partial without understanding limits imposed by noise. Unfortunately, mere description of noise grows exponentially with system size, becoming cumbersome even for modest sized systems of imminent practical interest. We fulfill the need for estimates on performing noisy quantum state preparation, verification, and observation. To do the estimation we propose fast numerical algorithms to maximize the expectation value of any $d$-dimensional observable over states of bounded purity. This bound on purity factors in noise in a measurable way. Our fastest algorithm takes $O(d)$ steps if the eigendecomposition of the observable is known, otherwise takes $O(d^3)$ steps at worst. The algorithms also solve maximum likelihood estimation for quantum state tomography with convex and even non-convex purity constraints. Numerics show performance of our key sub-routine (it finds in linear time a probability vector with bounded norm that most overlaps with a fixed vector) can be several orders of magnitude faster than a common state-of-the-art convex optimization solver. Our work fosters a practical way forward to asses limitations on QIP imposed by quantum noise. Along the way, we also give a simple but fundamental insight, noisy systems (equivalently noisy Hamiltonians) always give higher ground-state energy than their noiseless counterparts.
如果不了解噪声带来的限制,对实用量子信息处理(QIP)的评估仍然是片面的。遗憾的是,仅仅对噪声的描述会随着系统规模的扩大而呈指数增长,即使是对具有迫切实际意义的适度规模系统而言,也会变得非常麻烦。我们满足了对执行噪声量子态准备、验证和观测的估计需求。为了进行估算,我们提出了快速数值算法,以最大化纯度有界的状态上任何 $d$ 维观测值的期望值。这种纯度约束以可测量的方式将噪声因素考虑在内。如果已知观测值的特征分解,我们的最快算法需要 $O(d)$ 步,否则最差也需要 $O(d^3)$ 步。这些算法还解决了具有凸纯度约束甚至非凸纯度约束的量子态层析最大似然估计问题。数值计算表明,我们的关键子程序(它能在线性时间内找到与固定矢量重叠最多的有界规范概率矢量)的性能比最先进的普通凸优化求解器快几个数量级。我们的工作为评估量子噪声对 QIP 的限制提供了一条切实可行的途径。同时,我们还提出了一个简单而基本的见解:有噪声的系统(等同于有噪声的哈密顿)总是比无噪声的系统具有更高的基态能量。
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引用次数: 0
Non-Markovianity in High-Dimensional Open Quantum Systems using Next-generation Multicore Optical Fibers 使用下一代多芯光纤的高维开放量子系统中的非马尔可夫性
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-12 DOI: 10.22331/q-2024-08-12-1436
Santiago Rojas-Rojas, Daniel Martínez, Kei Sawada, Luciano Pereira, Stephen P. Walborn, Esteban S. Gómez, Nadja K. Bernardes, Gustavo Lima
With the advent of quantum technology, the interest in communication tasks assisted by quantum systems has increased both in academia and industry. Nonetheless, the transmission of a quantum state in real-world scenarios is bounded by environmental noise, so that the quantum channel is an open quantum system. In this work, we study a high-dimensional open quantum system in a multicore optical fiber by characterizing the environmental interaction as quantum operations corresponding to probabilistic phase-flips. The experimental platform is currently state-of-the-art for quantum information processing with multicore fibers. At a given evolution stage we observe a non-Markovian behaviour of the system, which is demonstrated through a proof-of-principle implementation of the Quantum Vault protocol. A better understanding of phase-noise in multicore fibers will improve several real-world communication protocols, since they are a prime candidate to be adopted in future telecom networks.
随着量子技术的出现,学术界和工业界对量子系统辅助通信任务的兴趣与日俱增。然而,量子态在现实世界中的传输受到环境噪声的限制,因此量子信道是一个开放的量子系统。在这项工作中,我们研究了多芯光纤中的高维开放量子系统,将环境相互作用表征为与概率相位翻转相对应的量子操作。该实验平台是目前使用多芯光纤进行量子信息处理的最先进平台。在给定的演化阶段,我们观察到系统的非马尔可夫行为,并通过量子金库协议的原理验证实施加以证明。更好地理解多核光纤中的相位噪声将改进现实世界中的一些通信协议,因为它们是未来电信网络采用的主要候选方案。
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引用次数: 0
Hayden-Preskill recovery in chaotic and integrable unitary circuit dynamics 混沌和可积分单元电路动力学中的海登-普雷斯基尔恢复
IF 6.4 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Pub Date : 2024-08-08 DOI: 10.22331/q-2024-08-08-1434
Michael A. Rampp, Pieter W. Claeys
The Hayden-Preskill protocol probes the capability of information recovery from local subsystems after unitary dynamics. As such it resolves the capability of quantum many-body systems to dynamically implement a quantum error-correcting code. The transition to coding behavior has been mostly discussed using effective approaches, such as entanglement membrane theory. Here, we present exact results on the use of Hayden-Preskill recovery as a dynamical probe of scrambling in local quantum many-body systems. We investigate certain classes of unitary circuit models, both structured Floquet (dual-unitary) and Haar-random circuits. We discuss different dynamical signatures corresponding to information transport or scrambling, respectively, that go beyond effective approaches. Surprisingly, certain chaotic circuits transport information with perfect fidelity. In integrable dual-unitary circuits, we relate the information transmission to the propagation and scattering of quasiparticles. Using numerical and analytical insights, we argue that the qualitative features of information recovery extend away from these solvable points. Our results suggest that information recovery protocols can serve to distinguish chaotic and integrable behavior, and that they are sensitive to characteristic dynamical features, such as long-lived quasiparticles or dual-unitarity.
海登-普雷斯基尔协议探究了单元动力学后从局部子系统恢复信息的能力。因此,它解决了量子多体系统动态实现量子纠错码的能力问题。向编码行为的过渡主要是通过有效的方法来讨论的,比如纠缠膜理论。在这里,我们提出了使用海登-普雷斯基尔恢复作为局部量子多体系统扰码动态探测的精确结果。我们研究了几类单元电路模型,包括结构化 Floquet 电路(双单元电路)和哈氏随机电路。我们分别讨论了与信息传输或扰乱相对应的不同动力学特征,这些特征超越了有效的方法。令人惊讶的是,某些混沌电路能完全保真地传输信息。在可积分双单元电路中,我们将信息传输与准粒子的传播和散射联系起来。通过数值和分析,我们认为信息恢复的定性特征可以从这些可解点延伸开来。我们的研究结果表明,信息恢复协议可用于区分混沌行为和可积分行为,而且对长寿命准粒子或双单调性等动态特征非常敏感。
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引用次数: 0
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