Pub Date : 2024-09-09DOI: 10.22331/q-2024-09-09-1464
Jacopo Surace
What would be the consequences if there were fundamental limits to our ability to experimentally explore the world? In this work we seriously consider this question. We assume the existence of statements whose truth value is not experimentally accessible. That is, there is no way, not even in theory, to directly test if these statements are true or false. We further develop a theory in which experimentally accessible statements are a union of a fixed minimum number of inaccessible statements. For example, the value of truth of the statements "a" and "b" is not accessible, but the value of truth of the statement "a or b" is accessible. We do not directly assume probability theory, we exclusively define experimentally accessible and inaccessible statements and build on these notions using the rules of classical logic. We find that an interesting structure emerges. Developing this theory, we relax the logical structure to a probabilistic one, obtaining a theory rich in structure that we call "theory of inaccessible information". Surprisingly, the simplest model of theory of inaccessible information is the qubit in quantum mechanics. Along the path for the construction of this theory, we characterise and study a family of multiplicative information measures that we call "inaccessibility measures".
如果我们在实验中探索世界的能力受到根本限制，会产生什么后果？在这项工作中，我们认真考虑了这个问题。我们假设存在一些无法通过实验获得其真值的语句。也就是说，我们甚至在理论上都无法直接检验这些语句的真假。我们进一步发展了一种理论，在这种理论中，实验上可获取的语句是固定的最少数量的不可获取语句的结合。例如，"a "和 "b "语句的真值是不可获取的，但 "a 或 b "语句的真值是可获取的。我们并不直接假设概率论，我们只定义了实验中可访问和不可访问的语句，并使用经典逻辑规则在这些概念的基础上进行构建。我们发现出现了一个有趣的结构。在发展这一理论的过程中，我们将逻辑结构放宽为概率结构，从而获得了一种结构丰富的理论，我们称之为 "不可获取信息理论"。令人惊讶的是，不可获取信息理论的最简单模型就是量子力学中的量子比特。在构建这一理论的过程中，我们描述并研究了一系列乘法信息度量，我们称之为 "不可获取度量"。
{"title":"A Theory of Inaccessible Information","authors":"Jacopo Surace","doi":"10.22331/q-2024-09-09-1464","DOIUrl":"https://doi.org/10.22331/q-2024-09-09-1464","url":null,"abstract":"What would be the consequences if there were fundamental limits to our ability to experimentally explore the world? In this work we seriously consider this question. We assume the existence of statements whose truth value is not experimentally accessible. That is, there is no way, not even in theory, to directly test if these statements are true or false. We further develop a theory in which experimentally accessible statements are a union of a fixed minimum number of inaccessible statements. For example, the value of truth of the statements \"a\" and \"b\" is not accessible, but the value of truth of the statement \"a or b\" is accessible. We do not directly assume probability theory, we exclusively define experimentally accessible and inaccessible statements and build on these notions using the rules of classical logic. We find that an interesting structure emerges. Developing this theory, we relax the logical structure to a probabilistic one, obtaining a theory rich in structure that we call \"theory of inaccessible information\". Surprisingly, the simplest model of theory of inaccessible information is the qubit in quantum mechanics. Along the path for the construction of this theory, we characterise and study a family of multiplicative information measures that we call \"inaccessibility measures\".","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"2 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142158686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-09DOI: 10.22331/q-2024-09-09-1466
A. Parra-Rodriguez, I. L. Egusquiza
In lumped-element electrical circuit theory, the problem of solving Maxwell's equations in the presence of media is reduced to two sets of equations, the constitutive equations encapsulating local geometry and dynamics of a confined energy density, and the Kirchhoff equations enforcing conservation of charge and energy in a larger, topological, scale. We develop a new geometric and systematic description of the dynamics of general lumped-element electrical circuits as first order differential equations, derivable from a Lagrangian and a Rayleigh dissipation function. Through the Faddeev-Jackiw method we identify and classify the singularities that arise in the search for Hamiltonian descriptions of general networks. The core of our solution relies on the correct identification of the reduced manifold in which the circuit state is expressible, e.g., a mix of flux and charge degrees of freedom, including the presence of compact ones. We apply our fully programmable method to obtain (canonically quantizable) Hamiltonian descriptions of nonlinear and nonreciprocal circuits which would be cumbersome/singular if pure node-flux or loop-charge variables were used as a starting configuration space. We also propose a specific assignment of topology for the branch variables of energetic elements, that when used as input to the procedure gives results consistent with classical descriptions as well as with spectra of more involved quantum circuits. This work unifies diverse existent geometrical pictures of electrical network theory, and will prove useful, for instance, to automatize the computation of exact Hamiltonian descriptions of superconducting quantum chips.
{"title":"Geometrical description and Faddeev-Jackiw quantization of electrical networks","authors":"A. Parra-Rodriguez, I. L. Egusquiza","doi":"10.22331/q-2024-09-09-1466","DOIUrl":"https://doi.org/10.22331/q-2024-09-09-1466","url":null,"abstract":"In lumped-element electrical circuit theory, the problem of solving Maxwell's equations in the presence of media is reduced to two sets of equations, the constitutive equations encapsulating local geometry and dynamics of a confined energy density, and the Kirchhoff equations enforcing conservation of charge and energy in a larger, topological, scale. We develop a new geometric and systematic description of the dynamics of general lumped-element electrical circuits as first order differential equations, derivable from a Lagrangian and a Rayleigh dissipation function. Through the Faddeev-Jackiw method we identify and classify the singularities that arise in the search for Hamiltonian descriptions of general networks. The core of our solution relies on the correct identification of the reduced manifold in which the circuit state is expressible, e.g., a mix of flux and charge degrees of freedom, including the presence of compact ones. We apply our fully programmable method to obtain (canonically quantizable) Hamiltonian descriptions of nonlinear and nonreciprocal circuits which would be cumbersome/singular if pure node-flux or loop-charge variables were used as a starting configuration space. We also propose a specific assignment of topology for the branch variables of energetic elements, that when used as input to the procedure gives results consistent with classical descriptions as well as with spectra of more involved quantum circuits. This work unifies diverse existent geometrical pictures of electrical network theory, and will prove useful, for instance, to automatize the computation of exact Hamiltonian descriptions of superconducting quantum chips.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"15 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142158688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-06DOI: 10.22331/q-2024-09-06-1462
Brian J. J. Khor, D. M. Kürkçüoglu, T. J. Hobbs, G. N. Perdue, Israel Klich
In this work, we explore the interplay of confinement, string breaking and entanglement asymmetry on a 1D quantum Ising chain. We consider the evolution of an initial domain wall and show that, surprisingly, while the introduction of confinement through a longitudinal field typically suppresses entanglement, it can also serve to increase it beyond a bound set for free particles. Our model can be tuned to conserve the number of domain walls, which gives an opportunity to explore entanglement asymmetry associated with link variables. We study two approaches to deal with the non-locality of the link variables, either directly or following a Kramers-Wannier transformation that maps bond variables (kinks) to site variables (spins). We develop a numerical procedure for computing the asymmetry using tensor network methods and use it to demonstrate the different types of entanglement and entanglement asymmetry.
{"title":"Confinement and Kink Entanglement Asymmetry on a Quantum Ising Chain","authors":"Brian J. J. Khor, D. M. Kürkçüoglu, T. J. Hobbs, G. N. Perdue, Israel Klich","doi":"10.22331/q-2024-09-06-1462","DOIUrl":"https://doi.org/10.22331/q-2024-09-06-1462","url":null,"abstract":"In this work, we explore the interplay of confinement, string breaking and entanglement asymmetry on a 1D quantum Ising chain. We consider the evolution of an initial domain wall and show that, surprisingly, while the introduction of confinement through a longitudinal field typically suppresses entanglement, it can also serve to increase it beyond a bound set for free particles. Our model can be tuned to conserve the number of domain walls, which gives an opportunity to explore entanglement asymmetry associated with link variables. We study two approaches to deal with the non-locality of the link variables, either directly or following a Kramers-Wannier transformation that maps bond variables (kinks) to site variables (spins). We develop a numerical procedure for computing the asymmetry using tensor network methods and use it to demonstrate the different types of entanglement and entanglement asymmetry.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"7 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142142714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-05DOI: 10.22331/q-2024-09-05-1461
Hiroki Hamaguchi, Kou Hamada, Nobuyuki Yoshioka
The nonstabilizerness, or magic, is an essential quantum resource to perform universal quantum computation. Robustness of magic (RoM) in particular characterizes the degree of usefulness of a given quantum state for non-Clifford operation. While the mathematical formalism of RoM can be given in a concise manner, it is extremely challenging to determine the RoM in practice, since it involves superexponentially many pure stabilizer states. In this work, we present efficient novel algorithms to compute the RoM. The crucial technique is a subroutine that achieves the remarkable features in calculation of overlaps between pure stabilizer states: (i) the time complexity per each stabilizer is reduced exponentially, (ii) the space complexity is reduced superexponentially. Based on this subroutine, we present algorithms to compute the RoM for arbitrary states up to $n=7$ qubits on a laptop, while brute-force methods require a memory size of 86 TiB. As a byproduct, the proposed subroutine allows us to simulate the stabilizer fidelity up to $n=8$ qubits, for which naive methods require memory size of 86 PiB so that any state-of-the-art classical computer cannot execute the computation. We further propose novel algorithms that utilize the preknowledge on the structure of target quantum state such as the permutation symmetry of disentanglement, and numerically demonstrate our state-of-the-art results for copies of magic states and partially disentangled quantum states. The series of algorithms constitute a comprehensive “handbook'' to scale up the computation of the RoM, and we envision that the proposed technique applies to the computation of other quantum resource measures as well.
不稳定性或魔力是执行通用量子计算的重要量子资源。魔力的鲁棒性（RoM）特别表征了给定量子态在非克里福德操作中的有用程度。虽然 RoM 的数学形式可以简明地给出，但在实践中确定 RoM 却极具挑战性，因为它涉及超指数的许多纯稳定态。在这项工作中，我们提出了计算 RoM 的高效新算法。其中的关键技术是一个子程序，它在计算纯稳定器状态之间的重叠时实现了以下显著特点：(i) 每个稳定器的时间复杂度呈指数级降低，(ii) 空间复杂度呈超指数级降低。基于这个子程序，我们提出了在笔记本电脑上计算高达 $n=7$ 量子比特的任意状态的 RoM 算法，而蛮力方法需要 86 TiB 的内存大小。作为副产品，我们提出的子程序允许我们模拟高达 $n=8$ 量子位的稳定器保真度，而对于这种稳定器保真度，天真方法需要 86 PiB 的内存容量，因此任何最先进的经典计算机都无法执行计算。我们进一步提出了利用目标量子态结构预知（如解纠缠的置换对称性）的新算法，并在魔态副本和部分解纠缠量子态上用数值证明了我们最先进的结果。这一系列算法构成了一本扩展 RoM 计算的综合 "手册"，我们设想所提出的技术也适用于其他量子资源度量的计算。
{"title":"Handbook for Efficiently Quantifying Robustness of Magic","authors":"Hiroki Hamaguchi, Kou Hamada, Nobuyuki Yoshioka","doi":"10.22331/q-2024-09-05-1461","DOIUrl":"https://doi.org/10.22331/q-2024-09-05-1461","url":null,"abstract":"The nonstabilizerness, or magic, is an essential quantum resource to perform universal quantum computation. Robustness of magic (RoM) in particular characterizes the degree of usefulness of a given quantum state for non-Clifford operation. While the mathematical formalism of RoM can be given in a concise manner, it is extremely challenging to determine the RoM in practice, since it involves superexponentially many pure stabilizer states. In this work, we present efficient novel algorithms to compute the RoM. The crucial technique is a subroutine that achieves the remarkable features in calculation of overlaps between pure stabilizer states: (i) the time complexity per each stabilizer is reduced exponentially, (ii) the space complexity is reduced superexponentially. Based on this subroutine, we present algorithms to compute the RoM for arbitrary states up to $n=7$ qubits on a laptop, while brute-force methods require a memory size of 86 TiB. As a byproduct, the proposed subroutine allows us to simulate the stabilizer fidelity up to $n=8$ qubits, for which naive methods require memory size of 86 PiB so that any state-of-the-art classical computer cannot execute the computation. We further propose novel algorithms that utilize the preknowledge on the structure of target quantum state such as the permutation symmetry of disentanglement, and numerically demonstrate our state-of-the-art results for copies of magic states and partially disentangled quantum states. The series of algorithms constitute a comprehensive “handbook'' to scale up the computation of the RoM, and we envision that the proposed technique applies to the computation of other quantum resource measures as well.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"20 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142138383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-04DOI: 10.22331/q-2024-09-04-1460
Marco Ballarin, Giovanni Cataldi, Giuseppe Magnifico, Daniel Jaschke, Marco Di Liberto, Ilaria Siloi, Simone Montangero, Pietro Silvi
We numerically analyze the feasibility of a platform-neutral, general strategy to perform quantum simulations of fermionic lattice field theories under open boundary conditions. The digital quantum simulator requires solely one- and two-qubit gates and is scalable since integrating each Hamiltonian term requires a finite (non-scaling) cost. The exact local fermion encoding we adopt relies on auxiliary $mathbb{Z}_2$ lattice gauge fields by adding a pure gauge Hamiltonian term akin to the Toric Code. By numerically emulating the quantum simulator real-time dynamics, we observe a timescale separation for spin- and charge-excitations in a spin-$frac{1}{2}$ Hubbard ladder in the $t-J$ model limit.
{"title":"Digital quantum simulation of lattice fermion theories with local encoding","authors":"Marco Ballarin, Giovanni Cataldi, Giuseppe Magnifico, Daniel Jaschke, Marco Di Liberto, Ilaria Siloi, Simone Montangero, Pietro Silvi","doi":"10.22331/q-2024-09-04-1460","DOIUrl":"https://doi.org/10.22331/q-2024-09-04-1460","url":null,"abstract":"We numerically analyze the feasibility of a platform-neutral, general strategy to perform quantum simulations of fermionic lattice field theories under open boundary conditions. The digital quantum simulator requires solely one- and two-qubit gates and is scalable since integrating each Hamiltonian term requires a finite (non-scaling) cost. The exact local fermion encoding we adopt relies on auxiliary $mathbb{Z}_2$ lattice gauge fields by adding a pure gauge Hamiltonian term akin to the Toric Code. By numerically emulating the quantum simulator real-time dynamics, we observe a timescale separation for spin- and charge-excitations in a spin-$frac{1}{2}$ Hubbard ladder in the $t-J$ model limit.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"7 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142131019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-03DOI: 10.22331/q-2024-09-03-1459
Michał Piotrak, Marek Kopciuch, Arash Dezhang Fard, Magdalena Smolis, Szymon Pustelny, Kamil Korzekwa
In this paper we introduce and investigate the concept of a $textit{perfect quantum protractor}$, a pure quantum state $|psirangleinmathcal{H}$ that generates three different orthogonal bases of $mathcal{H}$ under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variance-based measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a well-defined total angular momentum $j$, and we prove that they do not exist for $jin{1/2,2,5/2}$, but they do exist for $jin{1,3/2,3}$ (with numerical evidence for their existence when $j=7/2$). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not $textit{a priori}$ known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium-87, where we prepare a perfect quantum protractor for a spin-1 system, let it precess around $x$, $y$ or $z$ axis, and then employ it to optimally estimate the rotation angle.
{"title":"Perfect quantum protractors","authors":"Michał Piotrak, Marek Kopciuch, Arash Dezhang Fard, Magdalena Smolis, Szymon Pustelny, Kamil Korzekwa","doi":"10.22331/q-2024-09-03-1459","DOIUrl":"https://doi.org/10.22331/q-2024-09-03-1459","url":null,"abstract":"In this paper we introduce and investigate the concept of a $textit{perfect quantum protractor}$, a pure quantum state $|psirangleinmathcal{H}$ that generates three different orthogonal bases of $mathcal{H}$ under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variance-based measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a well-defined total angular momentum $j$, and we prove that they do not exist for $jin{1/2,2,5/2}$, but they do exist for $jin{1,3/2,3}$ (with numerical evidence for their existence when $j=7/2$). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not $textit{a priori}$ known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium-87, where we prepare a perfect quantum protractor for a spin-1 system, let it precess around $x$, $y$ or $z$ axis, and then employ it to optimally estimate the rotation angle.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"51 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142123503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-03DOI: 10.22331/q-2024-09-03-1458
Bethany Davies, Álvaro G. Iñesta, Stephanie Wehner
Quantum networks crucially rely on the availability of high-quality entangled pairs of qubits, known as entangled links, distributed across distant nodes. Maintaining the quality of these links is a challenging task due to the presence of time-dependent noise, also known as decoherence. Entanglement purification protocols offer a solution by converting multiple low-quality entangled states into a smaller number of higher-quality ones. In this work, we introduce a framework to analyse the performance of entanglement buffering setups that combine entanglement consumption, decoherence, and entanglement purification. We propose two key metrics: the availability, which is the steady-state probability that an entangled link is present, and the average consumed fidelity, which quantifies the steady-state quality of consumed links. We then investigate a two-node system, where each node possesses two quantum memories: one for long-term entanglement storage, and another for entanglement generation. We model this setup as a continuous-time stochastic process and derive analytical expressions for the performance metrics. Our findings unveil a trade-off between the availability and the average consumed fidelity. We also bound these performance metrics for a buffering system that employs the well-known bilocal Clifford purification protocols. Importantly, our analysis demonstrates that, in the presence of noise, consistently purifying the buffered entanglement increases the average consumed fidelity, even when some buffered entanglement is discarded due to purification failures.
{"title":"Entanglement buffering with two quantum memories","authors":"Bethany Davies, Álvaro G. Iñesta, Stephanie Wehner","doi":"10.22331/q-2024-09-03-1458","DOIUrl":"https://doi.org/10.22331/q-2024-09-03-1458","url":null,"abstract":"Quantum networks crucially rely on the availability of high-quality entangled pairs of qubits, known as entangled links, distributed across distant nodes. Maintaining the quality of these links is a challenging task due to the presence of time-dependent noise, also known as decoherence. Entanglement purification protocols offer a solution by converting multiple low-quality entangled states into a smaller number of higher-quality ones. In this work, we introduce a framework to analyse the performance of entanglement buffering setups that combine entanglement consumption, decoherence, and entanglement purification. We propose two key metrics: the availability, which is the steady-state probability that an entangled link is present, and the average consumed fidelity, which quantifies the steady-state quality of consumed links. We then investigate a two-node system, where each node possesses two quantum memories: one for long-term entanglement storage, and another for entanglement generation. We model this setup as a continuous-time stochastic process and derive analytical expressions for the performance metrics. Our findings unveil a trade-off between the availability and the average consumed fidelity. We also bound these performance metrics for a buffering system that employs the well-known bilocal Clifford purification protocols. Importantly, our analysis demonstrates that, in the presence of noise, consistently purifying the buffered entanglement increases the average consumed fidelity, even when some buffered entanglement is discarded due to purification failures.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"90 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142123502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-29DOI: 10.22331/q-2024-08-29-1454
C. Gonzalez-Ballestero
Most quantum theorists are familiar with different ways of describing the effective quantum dynamics of a system coupled to external degrees of freedom, such as the Born-Markov master equation or the adiabatic elimination. Understanding the deep connection between these -- sometimes apparently unrelated -- methods can be a powerful tool, allowing us to derive effective dynamics in unconventional systems or regimes. This tutorial aims at providing quantum theorists across multiple fields (e.g., quantum and atom optics, optomechanics, or hybrid quantum systems) with a self-contained practical toolbox to derive effective quantum dynamics, applicable to systems ranging from $N$-level emitters to mechanical resonators. First, we summarize the projector approach to open quantum systems and the derivation of the fundamental Nakajima-Zwanzig equation. Then, we show how three common effective equations, namely the Brownian master equation, the Born-Markov master equation, and the adiabatic elimination used in atom and molecular optics, can be derived from different perturbative expansions of the Nakajima-Zwanzig equation. We also solve in detail four specific examples using this formalism, namely a harmonic oscillator subject to displacement noise, the effective equations of a mechanical resonator cooled by an optical cavity, the Purcell effect for a qubit coupled to an optical cavity, and the adiabatic elimination in a Lambda system.
{"title":"Tutorial: projector approach to master equations for open quantum systems","authors":"C. Gonzalez-Ballestero","doi":"10.22331/q-2024-08-29-1454","DOIUrl":"https://doi.org/10.22331/q-2024-08-29-1454","url":null,"abstract":"Most quantum theorists are familiar with different ways of describing the effective quantum dynamics of a system coupled to external degrees of freedom, such as the Born-Markov master equation or the adiabatic elimination. Understanding the deep connection between these -- sometimes apparently unrelated -- methods can be a powerful tool, allowing us to derive effective dynamics in unconventional systems or regimes. This tutorial aims at providing quantum theorists across multiple fields (e.g., quantum and atom optics, optomechanics, or hybrid quantum systems) with a self-contained practical toolbox to derive effective quantum dynamics, applicable to systems ranging from $N$-level emitters to mechanical resonators. First, we summarize the projector approach to open quantum systems and the derivation of the fundamental Nakajima-Zwanzig equation. Then, we show how three common effective equations, namely the Brownian master equation, the Born-Markov master equation, and the adiabatic elimination used in atom and molecular optics, can be derived from different perturbative expansions of the Nakajima-Zwanzig equation. We also solve in detail four specific examples using this formalism, namely a harmonic oscillator subject to displacement noise, the effective equations of a mechanical resonator cooled by an optical cavity, the Purcell effect for a qubit coupled to an optical cavity, and the adiabatic elimination in a Lambda system.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"127 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-29DOI: 10.22331/q-2024-08-29-1457
William Kirby
This work provides a nonasymptotic error analysis of quantum Krylov algorithms based on real-time evolutions, subject to generic errors in the outputs of the quantum circuits. We prove upper and lower bounds on the resulting ground state energy estimates, and the error associated to the upper bound is linear in the input error rates. This resolves a misalignment between known numerics, which exhibit approximately linear error scaling, and prior theoretical analysis, which only provably obtained scaling with the error rate to the power $frac{2}{3}$. Our main technique is to express generic errors in terms of an effective target Hamiltonian studied in an effective Krylov space. These results provide a theoretical framework for understanding the main features of quantum Krylov errors.
{"title":"Analysis of quantum Krylov algorithms with errors","authors":"William Kirby","doi":"10.22331/q-2024-08-29-1457","DOIUrl":"https://doi.org/10.22331/q-2024-08-29-1457","url":null,"abstract":"This work provides a nonasymptotic error analysis of quantum Krylov algorithms based on real-time evolutions, subject to generic errors in the outputs of the quantum circuits. We prove upper and lower bounds on the resulting ground state energy estimates, and the error associated to the upper bound is linear in the input error rates. This resolves a misalignment between known numerics, which exhibit approximately linear error scaling, and prior theoretical analysis, which only provably obtained scaling with the error rate to the power $frac{2}{3}$. Our main technique is to express generic errors in terms of an effective target Hamiltonian studied in an effective Krylov space. These results provide a theoretical framework for understanding the main features of quantum Krylov errors.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"11 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-29DOI: 10.22331/q-2024-08-29-1456
Naga Dileep Varikuti, Soumik Bandyopadhyay
Quantum state designs, by enabling an efficient sampling of random quantum states, play a quintessential role in devising and benchmarking various quantum protocols with broad applications ranging from circuit designs to black hole physics. Symmetries, on the other hand, are expected to reduce the randomness of a state. Despite being ubiquitous, the effects of symmetry on quantum state designs remain an outstanding question. The recently introduced projected ensemble framework generates efficient approximate state $t$-designs by hinging on projective measurements and many-body quantum chaos. In this work, we examine the emergence of state designs from the random generator states exhibiting symmetries. Leveraging on translation symmetry, we analytically establish a sufficient condition for the measurement basis leading to the state $t$-designs. Then, by making use of the trace distance measure, we numerically investigate the convergence to the designs. Subsequently, we inspect the violation of the sufficient condition to identify bases that fail to converge. We further demonstrate the emergence of state designs in a physical system by studying the dynamics of a chaotic tilted field Ising chain with translation symmetry. We find faster convergence of the trace distance during the early time evolution in comparison to the cases when the symmetry is broken. To delineate the general applicability of our results, we extend our analysis to other symmetries. We expect our findings to pave the way for further exploration of deep thermalization and equilibration of closed and open quantum many-body systems.
{"title":"Unraveling the emergence of quantum state designs in systems with symmetry","authors":"Naga Dileep Varikuti, Soumik Bandyopadhyay","doi":"10.22331/q-2024-08-29-1456","DOIUrl":"https://doi.org/10.22331/q-2024-08-29-1456","url":null,"abstract":"Quantum state designs, by enabling an efficient sampling of random quantum states, play a quintessential role in devising and benchmarking various quantum protocols with broad applications ranging from circuit designs to black hole physics. Symmetries, on the other hand, are expected to reduce the randomness of a state. Despite being ubiquitous, the effects of symmetry on quantum state designs remain an outstanding question. The recently introduced projected ensemble framework generates efficient approximate state $t$-designs by hinging on projective measurements and many-body quantum chaos. In this work, we examine the emergence of state designs from the random generator states exhibiting symmetries. Leveraging on translation symmetry, we analytically establish a sufficient condition for the measurement basis leading to the state $t$-designs. Then, by making use of the trace distance measure, we numerically investigate the convergence to the designs. Subsequently, we inspect the violation of the sufficient condition to identify bases that fail to converge. We further demonstrate the emergence of state designs in a physical system by studying the dynamics of a chaotic tilted field Ising chain with translation symmetry. We find faster convergence of the trace distance during the early time evolution in comparison to the cases when the symmetry is broken. To delineate the general applicability of our results, we extend our analysis to other symmetries. We expect our findings to pave the way for further exploration of deep thermalization and equilibration of closed and open quantum many-body systems.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"94 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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