Pub Date : 2024-08-29DOI: 10.22331/q-2024-08-29-1455
Zhihui Zhu, Joseph M. Lukens, Brian T. Kirby
Classical shadows (CS) offer a resource-efficient means to estimate quantum observables, circumventing the need for exhaustive state tomography. Here, we clarify and explore the connection between CS techniques and least squares (LS) and regularized least squares (RLS) methods commonly used in machine learning and data analysis. By formal identification of LS and RLS "shadows" completely analogous to those in CS – namely, point estimators calculated from the empirical frequencies of single measurements – we show that both RLS and CS can be viewed as regularizers for the underdetermined regime, replacing the pseudoinverse with invertible alternatives. Through numerical simulations, we evaluate RLS and CS from three distinct angles: the tradeoff in bias and variance, mismatch between the expected and actual measurement distributions, and the interplay between the number of measurements and number of shots per measurement. Compared to CS, RLS attains lower variance at the expense of bias, is robust to distribution mismatch, and is more sensitive to the number of shots for a fixed number of state copies – differences that can be understood from the distinct approaches taken to regularization. Conceptually, our integration of LS, RLS, and CS under a unifying "shadow" umbrella aids in advancing the overall picture of CS techniques, while practically our results highlight the tradeoffs intrinsic to these measurement approaches, illuminating the circumstances under which either RLS or CS would be preferred, such as unverified randomness for the former or unbiased estimation for the latter.
{"title":"On the connection between least squares, regularization, and classical shadows","authors":"Zhihui Zhu, Joseph M. Lukens, Brian T. Kirby","doi":"10.22331/q-2024-08-29-1455","DOIUrl":"https://doi.org/10.22331/q-2024-08-29-1455","url":null,"abstract":"Classical shadows (CS) offer a resource-efficient means to estimate quantum observables, circumventing the need for exhaustive state tomography. Here, we clarify and explore the connection between CS techniques and least squares (LS) and regularized least squares (RLS) methods commonly used in machine learning and data analysis. By formal identification of LS and RLS \"shadows\" completely analogous to those in CS – namely, point estimators calculated from the empirical frequencies of single measurements – we show that both RLS and CS can be viewed as regularizers for the underdetermined regime, replacing the pseudoinverse with invertible alternatives. Through numerical simulations, we evaluate RLS and CS from three distinct angles: the tradeoff in bias and variance, mismatch between the expected and actual measurement distributions, and the interplay between the number of measurements and number of shots per measurement. Compared to CS, RLS attains lower variance at the expense of bias, is robust to distribution mismatch, and is more sensitive to the number of shots for a fixed number of state copies – differences that can be understood from the distinct approaches taken to regularization. Conceptually, our integration of LS, RLS, and CS under a unifying \"shadow\" umbrella aids in advancing the overall picture of CS techniques, while practically our results highlight the tradeoffs intrinsic to these measurement approaches, illuminating the circumstances under which either RLS or CS would be preferred, such as unverified randomness for the former or unbiased estimation for the latter.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"58 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090079","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 : 2024-08-28DOI: 10.22331/q-2024-08-28-1453
Jie Lin, Benjamin MacLellan, Sobhan Ghanbari, Julie Belleville, Khuong Tran, Luc Robichaud, Roger G. Melko, Hoi-Kwong Lo, Piotr Roztocki
$tt{GraphiQ}$ is a versatile open-source framework for designing photonic graph state generation schemes, with a particular emphasis on photon-emitter hybrid circuits. Built in Python, GraphiQ consists of a suite of design tools, including multiple simulation backends and optimization methods. The library supports scheme optimization in the presence of circuit imperfections, as well as user-defined optimization goals. Our framework thus represents a valuable tool for the development of practical schemes adhering to experimentally-relevant constraints. As graph states are a key resource for measurement-based quantum computing, all-photonic quantum repeaters, and robust quantum metrology, among others, we envision GraphiQ's broad impact for advancing quantum technologies.
{"title":"GraphiQ: Quantum circuit design for photonic graph states","authors":"Jie Lin, Benjamin MacLellan, Sobhan Ghanbari, Julie Belleville, Khuong Tran, Luc Robichaud, Roger G. Melko, Hoi-Kwong Lo, Piotr Roztocki","doi":"10.22331/q-2024-08-28-1453","DOIUrl":"https://doi.org/10.22331/q-2024-08-28-1453","url":null,"abstract":"$tt{GraphiQ}$ is a versatile open-source framework for designing photonic graph state generation schemes, with a particular emphasis on photon-emitter hybrid circuits. Built in Python, GraphiQ consists of a suite of design tools, including multiple simulation backends and optimization methods. The library supports scheme optimization in the presence of circuit imperfections, as well as user-defined optimization goals. Our framework thus represents a valuable tool for the development of practical schemes adhering to experimentally-relevant constraints. As graph states are a key resource for measurement-based quantum computing, all-photonic quantum repeaters, and robust quantum metrology, among others, we envision GraphiQ's broad impact for advancing quantum technologies.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"21 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084817","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 : 2024-08-28DOI: 10.22331/q-2024-08-28-1451
Andrew Zhao, Nicholas C. Rubin
Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the $special$ orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.
{"title":"Expanding the reach of quantum optimization with fermionic embeddings","authors":"Andrew Zhao, Nicholas C. Rubin","doi":"10.22331/q-2024-08-28-1451","DOIUrl":"https://doi.org/10.22331/q-2024-08-28-1451","url":null,"abstract":"Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the $special$ orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"2 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084818","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 : 2024-08-28DOI: 10.22331/q-2024-08-28-1452
Santiago Llorens, Gael Sentís, Ramon Muñoz-Tapia
A source assumed to prepare a specified reference state sometimes prepares an anomalous one. We address the task of identifying these anomalous states in a series of $n$ preparations with $k$ anomalies. We analyze the minimum-error protocol and the zero-error (unambiguous) protocol and obtain closed expressions for the success probability when both reference and anomalous states are known to the observer and anomalies can appear equally likely in any position of the preparation series. We find the solution using results from association schemes theory, thus establishing a connection between graph theory and quantum hypothesis testing. In particular, we use the Johnson association scheme which arises naturally from the Gram matrix of this problem. We also study the regime of large $n$ and obtain the expression of the success probability that is non-vanishing. Finally, we address the case in which the observer is blind to the reference and the anomalous states. This scenario requires a universal protocol for which we prove that in the asymptotic limit, the success probability corresponds to the average of the known state scenario.
{"title":"Quantum multi-anomaly detection","authors":"Santiago Llorens, Gael Sentís, Ramon Muñoz-Tapia","doi":"10.22331/q-2024-08-28-1452","DOIUrl":"https://doi.org/10.22331/q-2024-08-28-1452","url":null,"abstract":"A source assumed to prepare a specified reference state sometimes prepares an anomalous one. We address the task of identifying these anomalous states in a series of $n$ preparations with $k$ anomalies. We analyze the minimum-error protocol and the zero-error (unambiguous) protocol and obtain closed expressions for the success probability when both reference and anomalous states are known to the observer and anomalies can appear equally likely in any position of the preparation series. We find the solution using results from association schemes theory, thus establishing a connection between graph theory and quantum hypothesis testing. In particular, we use the Johnson association scheme which arises naturally from the Gram matrix of this problem. We also study the regime of large $n$ and obtain the expression of the success probability that is non-vanishing. Finally, we address the case in which the observer is blind to the reference and the anomalous states. This scenario requires a universal protocol for which we prove that in the asymptotic limit, the success probability corresponds to the average of the known state scenario.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"48 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084876","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 : 2024-08-27DOI: 10.22331/q-2024-08-27-1445
Peter Brown, Hamza Fawzi, Omar Fawzi
The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.
{"title":"Device-independent lower bounds on the conditional von Neumann entropy","authors":"Peter Brown, Hamza Fawzi, Omar Fawzi","doi":"10.22331/q-2024-08-27-1445","DOIUrl":"https://doi.org/10.22331/q-2024-08-27-1445","url":null,"abstract":"The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"5 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084874","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 : 2024-08-27DOI: 10.22331/q-2024-08-27-1450
Nicholas Connolly, Vivien Londe, Anthony Leverrier, Nicolas Delfosse
We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the maximum likelihood decoder that can be implemented in $O(N^2)$ bit operations where $N$ is the length of the quantum code. A probabilistic version of this decoder can be implemented in $O(N^{1.5})$ bit operations.
{"title":"Fast erasure decoder for hypergraph product codes","authors":"Nicholas Connolly, Vivien Londe, Anthony Leverrier, Nicolas Delfosse","doi":"10.22331/q-2024-08-27-1450","DOIUrl":"https://doi.org/10.22331/q-2024-08-27-1450","url":null,"abstract":"We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the maximum likelihood decoder that can be implemented in $O(N^2)$ bit operations where $N$ is the length of the quantum code. A probabilistic version of this decoder can be implemented in $O(N^{1.5})$ bit operations.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"6 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084816","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 : 2024-08-27DOI: 10.22331/q-2024-08-27-1447
Davide Lonigro, Fattah Sakuldee, Łukasz Cywiński, Dariusz Chruściński, Piotr Szańkowski
The multitime probability distributions obtained by repeatedly probing a quantum system via the measurement of an observable generally violate Kolmogorov's consistency property. Therefore, one cannot interpret such distributions as the result of the sampling of a single trajectory. We show that, nonetheless, they do result from the sampling of one $pair$ of trajectories. In this sense, rather than give up on trajectories, quantum mechanics requires to double down on them. To this purpose, we prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions (that is, defined on pairs of elements of the original sample spaces), and we employ this result in the quantum mechanical scenario. We also discuss the relation of our results with the quantum comb formalism.
{"title":"Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics","authors":"Davide Lonigro, Fattah Sakuldee, Łukasz Cywiński, Dariusz Chruściński, Piotr Szańkowski","doi":"10.22331/q-2024-08-27-1447","DOIUrl":"https://doi.org/10.22331/q-2024-08-27-1447","url":null,"abstract":"The multitime probability distributions obtained by repeatedly probing a quantum system via the measurement of an observable generally violate Kolmogorov's consistency property. Therefore, one cannot interpret such distributions as the result of the sampling of a single trajectory. We show that, nonetheless, they do result from the sampling of one $pair$ of trajectories. In this sense, rather than give up on trajectories, quantum mechanics requires to double down on them. To this purpose, we prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions (that is, defined on pairs of elements of the original sample spaces), and we employ this result in the quantum mechanical scenario. We also discuss the relation of our results with the quantum comb formalism.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"12 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084875","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 : 2024-08-27DOI: 10.22331/q-2024-08-27-1449
Alexander Zlokapa, Rolando D. Somma
We consider the task of simulating time evolution under a Hamiltonian $H$ within its low-energy subspace. Assuming access to a block-encoding of $H':=(H-E)/lambda$, for some $lambda gt 0$ and $E in mathbb R$, the goal is to implement an $epsilon$-approximation to the evolution operator $e^{-itH}$ when the initial state is confined to the subspace corresponding to eigenvalues $[-1, -1+Delta/lambda]$ of $H'$, for $Delta leq lambda$. We present a quantum algorithm that requires $mathcal{O}(tsqrt{lambdaGamma} + sqrt{lambda/Gamma}log(1/epsilon))$ queries to the block-encoding for any choice of $Gamma$ such that $Delta leq Gamma leq lambda$. When the parameters satisfy $log(1/epsilon) = o(tlambda)$ and $Delta/lambda = o(1)$, this result improves over generic methods with query complexity $Omega(tlambda)$. Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform.
For a given $H$, the block-encoding of its $H'$ must be prepared efficiently to achieve an asymptotic speedup in simulating the low-energy subspace; we refer to these Hamiltonians as $gap-amplifiable$. We show necessary and sufficient conditions for gap amplifiability in terms of an operationally useful decomposition of $H$ into a sum of squares. Gap-amplifiable Hamiltonians include physically relevant examples such as frustration-free systems, and it encompasses all previously considered settings of low energy simulation algorithms. Any Hamiltonian can be expressed as a gap-amplifiable Hamiltonian after simple transformations, and our algorithm retains the asymptotic improvement over generic methods as long as the conditions on the parameters are met.
We also provide lower bounds for simulating dynamics of low-energy states. In the worst case, we show that the low-energy condition cannot be used to improve the runtime of Hamiltonian simulation methods. For gap-amplifiable Hamiltonians, we prove that our algorithm is tight in the query model with respect to $t$, $Delta$, and $lambda$. In the practically relevant regime where $log (1/epsilon) = o(tDelta)$ and $Delta/lambda = o(1)$, we also prove a matching lower bound in gate complexity (up to logarithmic factors). To establish the query lower bounds, we consider oracular problems including search and $mathrm{PARITY}circmathrm{OR}$, and also bounds on the degrees of trigonometric polynomials. To establish the lower bound on gate complexity, we use a circuit-to-Hamiltonian reduction, where a “clock Hamiltonian'' acting on a low-energy state can simulate any quantum circuit.
我们考虑的任务是在低能子空间内模拟哈密顿方程 $H$ 下的时间演化。假设可以获得 $H'：=(H-E)/lambda$, for some $lambda gt 0$ and $E in mathbb R$, the goal is to implement an $epsilon$-approximation to the evolution operator $e^{-itH}$ when the initial state is confined to the subspace corresponding to eigenvalues $[-1, -1+Delta/lambda]$ of $H'$, for $Delta leq lambda$.我们提出了一种量子算法，它需要 $mathcal{O}(tsqrt{lambdaGamma} + sqrt{/lambda//Gamma}log(1/epsilon))$对任意选择的$Gamma$进行块编码查询，使得$Delta leq Gamma leq lambda$。当参数满足$log(1/epsilon) = o(t/lambda)$和$Delta/lambda = o(1)$时，这个结果比查询复杂度为$Omega(t/lambda)$的一般方法要好。我们的量子算法利用了频谱间隙放大和量子奇异值变换。对于给定的 $H$，其 $H'$ 的块编码必须高效准备，以实现模拟低能子空间的渐近加速；我们把这些哈密顿称作 $gap-amplable$。我们通过将 $H$ 分解为一个平方和的实用操作方法，展示了间隙可放大性的必要条件和充分条件。间隙可放大哈密顿数包括与物理相关的例子，如无挫折系统，它涵盖了以前考虑过的所有低能量模拟算法设置。任何哈密顿都可以在简单变换后表示为可间隙放大哈密顿，只要参数条件满足，我们的算法就能保持对一般方法的渐进改进。我们还提供了模拟低能态动力学的下限。在最坏的情况下，我们证明低能条件不能用来改善哈密顿模拟方法的运行时间。对于间隙可放大的哈密顿，我们证明了我们的算法在查询模型中与 $t$、$Delta$ 和 $lambda$ 有关是紧密的。在$log (1/epsilon) = o(t/Delta)$和$Delta/lambda = o(1)$的实际相关机制中，我们还证明了门复杂度的匹配下限（达到对数因子）。为了建立查询下界，我们考虑了包括搜索和 $mathrm{PARITY}circmathrm{OR}$ 在内的奥拉格问题，以及三角多项式的度数下界。为了建立门复杂性的下界，我们使用了电路到哈密顿的还原，其中作用于低能态的 "时钟哈密顿 "可以模拟任何量子电路。
{"title":"Hamiltonian simulation for low-energy states with optimal time dependence","authors":"Alexander Zlokapa, Rolando D. Somma","doi":"10.22331/q-2024-08-27-1449","DOIUrl":"https://doi.org/10.22331/q-2024-08-27-1449","url":null,"abstract":"We consider the task of simulating time evolution under a Hamiltonian $H$ within its low-energy subspace. Assuming access to a block-encoding of $H':=(H-E)/lambda$, for some $lambda gt 0$ and $E in mathbb R$, the goal is to implement an $epsilon$-approximation to the evolution operator $e^{-itH}$ when the initial state is confined to the subspace corresponding to eigenvalues $[-1, -1+Delta/lambda]$ of $H'$, for $Delta leq lambda$. We present a quantum algorithm that requires $mathcal{O}(tsqrt{lambdaGamma} + sqrt{lambda/Gamma}log(1/epsilon))$ queries to the block-encoding for any choice of $Gamma$ such that $Delta leq Gamma leq lambda$. When the parameters satisfy $log(1/epsilon) = o(tlambda)$ and $Delta/lambda = o(1)$, this result improves over generic methods with query complexity $Omega(tlambda)$. Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform.<br/>\u0000<br/> For a given $H$, the block-encoding of its $H'$ must be prepared efficiently to achieve an asymptotic speedup in simulating the low-energy subspace; we refer to these Hamiltonians as $gap-amplifiable$. We show necessary and sufficient conditions for gap amplifiability in terms of an operationally useful decomposition of $H$ into a sum of squares. Gap-amplifiable Hamiltonians include physically relevant examples such as frustration-free systems, and it encompasses all previously considered settings of low energy simulation algorithms. Any Hamiltonian can be expressed as a gap-amplifiable Hamiltonian after simple transformations, and our algorithm retains the asymptotic improvement over generic methods as long as the conditions on the parameters are met.<br/>\u0000<br/> We also provide lower bounds for simulating dynamics of low-energy states. In the worst case, we show that the low-energy condition cannot be used to improve the runtime of Hamiltonian simulation methods. For gap-amplifiable Hamiltonians, we prove that our algorithm is tight in the query model with respect to $t$, $Delta$, and $lambda$. In the practically relevant regime where $log (1/epsilon) = o(tDelta)$ and $Delta/lambda = o(1)$, we also prove a matching lower bound in gate complexity (up to logarithmic factors). To establish the query lower bounds, we consider oracular problems including search and $mathrm{PARITY}circmathrm{OR}$, and also bounds on the degrees of trigonometric polynomials. To establish the lower bound on gate complexity, we use a circuit-to-Hamiltonian reduction, where a “clock Hamiltonian'' acting on a low-energy state can simulate any quantum circuit.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"29 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084815","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 : 2024-08-27DOI: 10.22331/q-2024-08-27-1448
Margarita Davydova, Nathanan Tantivasadakarn, Shankar Balasubramanian, David Aasen
We propose a new model of quantum computation comprised of low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply logical gates. These measurement sequences constitute a new class of quantum error-correcting codes generalizing Floquet codes, which we call dynamic automorphism (DA) codes. We construct an explicit example, the DA color code, which is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of $N$ triangular patches, the DA color code encodes $N$ logical qubits and can implement the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements. We also make the first step towards universal quantum computation with DA codes by introducing a 3D DA color code and showing that a non-Clifford logical gate can be realized by adaptive two-qubit measurements.
我们提出了一种新的量子计算模型，它由同时编码逻辑信息、实现纠错和应用逻辑门的低量测量序列组成。这些测量序列构成了一类新的量子纠错码，概括了 Floquet 码，我们称之为动态自动变形（DA）码。我们构建了一个明确的例子--DA 色码，它由短测量序列组合而成，可以实现二维色码的所有 72 个自动变形。在一个由 N$ 三角形补丁组成的堆栈上，DA 色码编码 N$ 逻辑量子比特，并能通过一个二量子比特序列（更罕见的是三量子比特保利测量序列）实现完整的逻辑克利福德群。我们还通过引入三维 DA 颜色码，证明非克利福德逻辑门可以通过自适应二量子比特测量来实现，从而向使用 DA 码进行通用量子计算迈出了第一步。
{"title":"Quantum computation from dynamic automorphism codes","authors":"Margarita Davydova, Nathanan Tantivasadakarn, Shankar Balasubramanian, David Aasen","doi":"10.22331/q-2024-08-27-1448","DOIUrl":"https://doi.org/10.22331/q-2024-08-27-1448","url":null,"abstract":"We propose a new model of quantum computation comprised of low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply logical gates. These measurement sequences constitute a new class of quantum error-correcting codes generalizing Floquet codes, which we call dynamic automorphism (DA) codes. We construct an explicit example, the DA color code, which is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of $N$ triangular patches, the DA color code encodes $N$ logical qubits and can implement the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements. We also make the first step towards universal quantum computation with DA codes by introducing a 3D DA color code and showing that a non-Clifford logical gate can be realized by adaptive two-qubit measurements.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"51 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084813","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 : 2024-08-27DOI: 10.22331/q-2024-08-27-1446
Apollonas S. Matsoukas-Roubeas, Tomaž Prosen, Adolfo del Campo
The survival probability of an initial Coherent Gibbs State (CGS) is a natural extension of the Spectral Form Factor (SFF) to open quantum systems. To quantify the interplay between quantum chaos and decoherence away from the semi-classical limit, we investigate the relation of this generalized SFF with the corresponding $l_1$-norm of coherence. As a working example, we introduce Parametric Quantum Channels (PQC), a discrete-time model of unitary evolution mixed with the effects of measurements or transient interactions with an environment. The Energy Dephasing (ED) dynamics arises as a specific case in the Markovian limit. We demonstrate our results in a series of random matrix models.
{"title":"Quantum Chaos and Coherence: Random Parametric Quantum Channels","authors":"Apollonas S. Matsoukas-Roubeas, Tomaž Prosen, Adolfo del Campo","doi":"10.22331/q-2024-08-27-1446","DOIUrl":"https://doi.org/10.22331/q-2024-08-27-1446","url":null,"abstract":"The survival probability of an initial Coherent Gibbs State (CGS) is a natural extension of the Spectral Form Factor (SFF) to open quantum systems. To quantify the interplay between quantum chaos and decoherence away from the semi-classical limit, we investigate the relation of this generalized SFF with the corresponding $l_1$-norm of coherence. As a working example, we introduce Parametric Quantum Channels (PQC), a discrete-time model of unitary evolution mixed with the effects of measurements or transient interactions with an environment. The Energy Dephasing (ED) dynamics arises as a specific case in the Markovian limit. We demonstrate our results in a series of random matrix models.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"71 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084820","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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