Pub Date : 2024-07-17DOI: 10.22331/q-2024-07-17-1415
Simon Milz, Marco Túlio Quintino
Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces. This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order. Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects. Here, we provide a general framework to deduce these properties in a direct and easy to use way. While primarily guided by practical quantum mechanical considerations, we also extend our analysis to mappings between $general$ linear/affine spaces and derive their properties, opening the possibility for analysing sets which are not explicitly forbidden by quantum theory, but are still not much explored. Together, these results yield versatile and readily applicable tools for all tasks that require the characterisation of linear transformations, in quantum mechanics and beyond. As an application of our methods, we discuss how the existence of indefinite causality naturally emerges in higher-order quantum transformations and provide a simple strategy for the characterisation of mappings that have to preserve properties in a 'complete' sense, i.e., when acting non-trivially only on parts of an input space.
{"title":"Characterising transformations between quantum objects, ‘completeness’ of quantum properties, and transformations without a fixed causal order","authors":"Simon Milz, Marco Túlio Quintino","doi":"10.22331/q-2024-07-17-1415","DOIUrl":"https://doi.org/10.22331/q-2024-07-17-1415","url":null,"abstract":"Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces. This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order. Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects. Here, we provide a general framework to deduce these properties in a direct and easy to use way. While primarily guided by practical quantum mechanical considerations, we also extend our analysis to mappings between $general$ linear/affine spaces and derive their properties, opening the possibility for analysing sets which are not explicitly forbidden by quantum theory, but are still not much explored. Together, these results yield versatile and readily applicable tools for all tasks that require the characterisation of linear transformations, in quantum mechanics and beyond. As an application of our methods, we discuss how the existence of indefinite causality naturally emerges in higher-order quantum transformations and provide a simple strategy for the characterisation of mappings that have to preserve properties in a 'complete' sense, i.e., when acting non-trivially only on parts of an input space.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141726007","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 : 2024-07-15DOI: 10.22331/q-2024-07-15-1409
Weiyuan Gong, Shuo Zhou, Tongyang Li
Digital quantum simulation has broad applications in approximating unitary evolution of Hamiltonians. In practice, many simulation tasks for quantum systems focus on quantum states in the low-energy subspace instead of the entire Hilbert space. In this paper, we systematically investigate the complexity of digital quantum simulation based on product formulas in the low-energy subspace. We show that the simulation error depends on the effective low-energy norm of the Hamiltonian for a variety of digital quantum simulation algorithms and quantum systems, allowing improvements over the previous complexities for full unitary simulations even for imperfect state preparations due to thermalization. In particular, for simulating spin models in the low-energy subspace, we prove that randomized product formulas such as qDRIFT and random permutation require smaller Trotter numbers. Such improvement also persists in symmetry-protected digital quantum simulations. We prove a similar improvement in simulating the dynamics of power-law quantum interactions. We also provide a query lower bound for general digital quantum simulations in the low-energy subspace.
{"title":"Complexity of Digital Quantum Simulation in the Low-Energy Subspace: Applications and a Lower Bound","authors":"Weiyuan Gong, Shuo Zhou, Tongyang Li","doi":"10.22331/q-2024-07-15-1409","DOIUrl":"https://doi.org/10.22331/q-2024-07-15-1409","url":null,"abstract":"Digital quantum simulation has broad applications in approximating unitary evolution of Hamiltonians. In practice, many simulation tasks for quantum systems focus on quantum states in the low-energy subspace instead of the entire Hilbert space. In this paper, we systematically investigate the complexity of digital quantum simulation based on product formulas in the low-energy subspace. We show that the simulation error depends on the effective low-energy norm of the Hamiltonian for a variety of digital quantum simulation algorithms and quantum systems, allowing improvements over the previous complexities for full unitary simulations even for imperfect state preparations due to thermalization. In particular, for simulating spin models in the low-energy subspace, we prove that randomized product formulas such as qDRIFT and random permutation require smaller Trotter numbers. Such improvement also persists in symmetry-protected digital quantum simulations. We prove a similar improvement in simulating the dynamics of power-law quantum interactions. We also provide a query lower bound for general digital quantum simulations in the low-energy subspace.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141618364","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 : 2024-07-15DOI: 10.22331/q-2024-07-15-1410
Samuele Ferracin, Akel Hashim, Jean-Loup Ville, Ravi Naik, Arnaud Carignan-Dugas, Hammam Qassim, Alexis Morvan, David I. Santiago, Irfan Siddiqi, Joel J. Wallman
Using near-term quantum computers to achieve a quantum advantage requires efficient strategies to improve the performance of the noisy quantum devices presently available. We develop and experimentally validate two efficient error mitigation protocols named ``Noiseless Output Extrapolation" and ``Pauli Error Cancellation" that can drastically enhance the performance of quantum circuits composed of noisy cycles of gates. By combining popular mitigation strategies such as probabilistic error cancellation and noise amplification with efficient noise reconstruction methods, our protocols can mitigate a wide range of noise processes that do not satisfy the assumptions underlying existing mitigation protocols, including non-local and gate-dependent processes. We test our protocols on a four-qubit superconducting processor at the Advanced Quantum Testbed. We observe significant improvements in the performance of both structured and random circuits, with up to $86%$ improvement in variation distance over the unmitigated outputs. Our experiments demonstrate the effectiveness of our protocols, as well as their practicality for current hardware platforms.
{"title":"Efficiently improving the performance of noisy quantum computers","authors":"Samuele Ferracin, Akel Hashim, Jean-Loup Ville, Ravi Naik, Arnaud Carignan-Dugas, Hammam Qassim, Alexis Morvan, David I. Santiago, Irfan Siddiqi, Joel J. Wallman","doi":"10.22331/q-2024-07-15-1410","DOIUrl":"https://doi.org/10.22331/q-2024-07-15-1410","url":null,"abstract":"Using near-term quantum computers to achieve a quantum advantage requires efficient strategies to improve the performance of the noisy quantum devices presently available. We develop and experimentally validate two efficient error mitigation protocols named ``Noiseless Output Extrapolation\" and ``Pauli Error Cancellation\" that can drastically enhance the performance of quantum circuits composed of noisy cycles of gates. By combining popular mitigation strategies such as probabilistic error cancellation and noise amplification with efficient noise reconstruction methods, our protocols can mitigate a wide range of noise processes that do not satisfy the assumptions underlying existing mitigation protocols, including non-local and gate-dependent processes. We test our protocols on a four-qubit superconducting processor at the Advanced Quantum Testbed. We observe significant improvements in the performance of both structured and random circuits, with up to $86%$ improvement in variation distance over the unmitigated outputs. Our experiments demonstrate the effectiveness of our protocols, as well as their practicality for current hardware platforms.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141618365","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 : 2024-07-15DOI: 10.22331/q-2024-07-15-1411
Pablo Viñas Martínez, Esperanza López, Alejandro Bermudez
The quantum simulation of magnetism in trapped-ion systems makes use of the crystal vibrations to mediate pairwise interactions between spins, which are encoded in the internal electronic states of the ions, and measured in experiments that probe the real-time dynamics. These interactions can be accounted for by a long-wavelength relativistic theory, where the phonons are described by a coarse-grained Klein-Gordon field $phi(x)$ locally coupled to the spins that acts as a carrier, leading to an analogue of pion-mediated Yukawa interactions. In the vicinity of a structural transition of the ion crystal, one must go beyond the Klein-Gordon fields, and include additional $lambdaphi^4$ terms responsible for phonon-phonon scattering. This leads to quantum effects that can be expressed by Feynman loop integrals that modify the range of the Yukawa-type spin interactions; an effect that could be used to probe the underlying fixed point of this quantum field theory (QFT). Unfortunately, the rigidity of the trapped-ion crystal makes it challenging to observe genuine quantum effects, such as the flow of the critical point with the quartic coupling $lambda$. We hereby show that thermal effects, which can be controlled by laser cooling, can unveil this flow through the appearance of thermal masses in interacting QFTs. We perform self-consistent calculations that resum certain Feynman diagrams and, additionally, go beyond mean-field theory to predict how measurements on the trapped-ion spin system can probe key properties of the $lambdaphi^4$ QFT.
{"title":"Thermal masses and trapped-ion quantum spin models: a self-consistent approach to Yukawa-type interactions in the $λ!ϕ^4$ model","authors":"Pablo Viñas Martínez, Esperanza López, Alejandro Bermudez","doi":"10.22331/q-2024-07-15-1411","DOIUrl":"https://doi.org/10.22331/q-2024-07-15-1411","url":null,"abstract":"The quantum simulation of magnetism in trapped-ion systems makes use of the crystal vibrations to mediate pairwise interactions between spins, which are encoded in the internal electronic states of the ions, and measured in experiments that probe the real-time dynamics. These interactions can be accounted for by a long-wavelength relativistic theory, where the phonons are described by a coarse-grained Klein-Gordon field $phi(x)$ locally coupled to the spins that acts as a carrier, leading to an analogue of pion-mediated Yukawa interactions. In the vicinity of a structural transition of the ion crystal, one must go beyond the Klein-Gordon fields, and include additional $lambdaphi^4$ terms responsible for phonon-phonon scattering. This leads to quantum effects that can be expressed by Feynman loop integrals that modify the range of the Yukawa-type spin interactions; an effect that could be used to probe the underlying fixed point of this quantum field theory (QFT). Unfortunately, the rigidity of the trapped-ion crystal makes it challenging to observe genuine quantum effects, such as the flow of the critical point with the quartic coupling $lambda$. We hereby show that thermal effects, which can be controlled by laser cooling, can unveil this flow through the appearance of thermal masses in interacting QFTs. We perform self-consistent calculations that resum certain Feynman diagrams and, additionally, go beyond mean-field theory to predict how measurements on the trapped-ion spin system can probe key properties of the $lambdaphi^4$ QFT.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141618366","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 : 2024-07-11DOI: 10.22331/q-2024-07-11-1407
Ke Wang, Xiantao Li
Models for open quantum systems, which play important roles in electron transport problems and quantum computing, must take into account the interaction of the quantum system with the surrounding environment. Although such models can be derived in some special cases, in most practical situations, the exact models are unknown and have to be calibrated. This paper presents a learning method to infer parameters in Markovian open quantum systems from measurement data. One important ingredient in the method is a direct simulation technique of the quantum master equation, which is designed to preserve the completely-positive property with guaranteed accuracy. The method is particularly helpful in the situation where the time intervals between measurements are large. The approach is validated with error estimates and numerical experiments.
{"title":"Simulation-assisted learning of open quantum systems","authors":"Ke Wang, Xiantao Li","doi":"10.22331/q-2024-07-11-1407","DOIUrl":"https://doi.org/10.22331/q-2024-07-11-1407","url":null,"abstract":"Models for open quantum systems, which play important roles in electron transport problems and quantum computing, must take into account the interaction of the quantum system with the surrounding environment. Although such models can be derived in some special cases, in most practical situations, the exact models are unknown and have to be calibrated. This paper presents a learning method to infer parameters in Markovian open quantum systems from measurement data. One important ingredient in the method is a direct simulation technique of the quantum master equation, which is designed to preserve the completely-positive property with guaranteed accuracy. The method is particularly helpful in the situation where the time intervals between measurements are large. The approach is validated with error estimates and numerical experiments.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584497","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 : 2024-07-11DOI: 10.22331/q-2024-07-11-1408
Jovan Jovanović, Carolin Wille, Daan Timmers, Steven H. Simon
In this work we present a proposal for realising non-Abelian anyons on a NISQ device. In particular we explore the feasibility of implementing the quantum double model $D(D_4)$. We propose techniques to drastically simplify the circuits for the manipulation and measurements of anyons. Numerical simulations with realistic noise models suggest that current NISQ technology is capable of probing signatures of non-Abelian anyons far beyond elemental properties such as the non-commutativity of braids. In particular, we conclude that experimentally measuring the full modular data of the model is feasible.
{"title":"A proposal to demonstrate non-abelian anyons on a NISQ device","authors":"Jovan Jovanović, Carolin Wille, Daan Timmers, Steven H. Simon","doi":"10.22331/q-2024-07-11-1408","DOIUrl":"https://doi.org/10.22331/q-2024-07-11-1408","url":null,"abstract":"In this work we present a proposal for realising non-Abelian anyons on a NISQ device. In particular we explore the feasibility of implementing the quantum double model $D(D_4)$. We propose techniques to drastically simplify the circuits for the manipulation and measurements of anyons. Numerical simulations with realistic noise models suggest that current NISQ technology is capable of probing signatures of non-Abelian anyons far beyond elemental properties such as the non-commutativity of braids. In particular, we conclude that experimentally measuring the full modular data of the model is feasible.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584216","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 : 2024-07-11DOI: 10.22331/q-2024-07-11-1406
Paolo Zanardi, Emanuel Dallas, Faidon Andreadakis, Seth Lloyd
In this paper we will attempt to answer the following question: what are the natural quantum subsystems which emerge out of a system's dynamical laws? To answer this question we first define generalized tensor product structures (gTPS) in terms of observables, as dual pairs of an operator subalgebra $cal A$ and its commutant. Second, we propose an operational criterion of minimal information scrambling at short time scales to dynamically select gTPS. In this way the emergent subsystems are those which maintain the longest informational identity. This strategy is made quantitative by defining a Gaussian scrambling rate in terms of the short-time expansion of an algebraic version of the Out of Time Order Correlation (OTOC) function i.e., the $cal A$-OTOC. The Gaussian scrambling rate is computed analytically for physically important cases of general division into subsystems, and is shown to have an intuitive and compelling physical interpretation in terms of minimizing the interaction strength between subsystems.
{"title":"Operational Quantum Mereology and Minimal Scrambling","authors":"Paolo Zanardi, Emanuel Dallas, Faidon Andreadakis, Seth Lloyd","doi":"10.22331/q-2024-07-11-1406","DOIUrl":"https://doi.org/10.22331/q-2024-07-11-1406","url":null,"abstract":"In this paper we will attempt to answer the following question: what are the natural quantum subsystems which emerge out of a system's dynamical laws? To answer this question we first define generalized tensor product structures (gTPS) in terms of observables, as dual pairs of an operator subalgebra $cal A$ and its commutant. Second, we propose an operational criterion of minimal information scrambling at short time scales to dynamically select gTPS. In this way the emergent subsystems are those which maintain the longest informational identity. This strategy is made quantitative by defining a Gaussian scrambling rate in terms of the short-time expansion of an algebraic version of the Out of Time Order Correlation (OTOC) function i.e., the $cal A$-OTOC. The Gaussian scrambling rate is computed analytically for physically important cases of general division into subsystems, and is shown to have an intuitive and compelling physical interpretation in terms of minimizing the interaction strength between subsystems.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584215","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 : 2024-07-10DOI: 10.22331/q-2024-07-10-1402
Pierre Botteron, Anne Broadbent, Reda Chhaibi, Ion Nechita, Clément Pellegrini
Communication complexity quantifies how difficult it is for two distant computers to evaluate a function $f(X,Y)$, where the strings $X$ and $Y$ are distributed to the first and second computer respectively, under the constraint of exchanging as few bits as possible. Surprisingly, some nonlocal boxes, which are resources shared by the two computers, are so powerful that they allow to $collapse$ communication complexity, in the sense that any Boolean function f can be correctly estimated with the exchange of only one bit of communication. The Popescu-Rohrlich (PR) box is an example of such a collapsing resource, but a comprehensive description of the set of collapsing nonlocal boxes remains elusive.
In this work, we carry out an algebraic study of the structure of wirings connecting nonlocal boxes, thus defining the notion of the "product of boxes" $Pboxtimes Q$, and we show related associativity and commutativity results. This gives rise to the notion of the "orbit of a box", unveiling surprising geometrical properties about the alignment and parallelism of distilled boxes. The power of this new framework is that it allows us to prove previously-reported numerical observations concerning the best way to wire consecutive boxes, and to numerically and analytically recover recently-identified noisy $texttt{PR}$ boxes that collapse communication complexity for different types of noise models.
{"title":"Algebra of Nonlocal Boxes and the Collapse of Communication Complexity","authors":"Pierre Botteron, Anne Broadbent, Reda Chhaibi, Ion Nechita, Clément Pellegrini","doi":"10.22331/q-2024-07-10-1402","DOIUrl":"https://doi.org/10.22331/q-2024-07-10-1402","url":null,"abstract":"Communication complexity quantifies how difficult it is for two distant computers to evaluate a function $f(X,Y)$, where the strings $X$ and $Y$ are distributed to the first and second computer respectively, under the constraint of exchanging as few bits as possible. Surprisingly, some nonlocal boxes, which are resources shared by the two computers, are so powerful that they allow to $collapse$ communication complexity, in the sense that any Boolean function f can be correctly estimated with the exchange of only one bit of communication. The Popescu-Rohrlich (PR) box is an example of such a collapsing resource, but a comprehensive description of the set of collapsing nonlocal boxes remains elusive.<br/>\u0000<br/> In this work, we carry out an algebraic study of the structure of wirings connecting nonlocal boxes, thus defining the notion of the \"product of boxes\" $Pboxtimes Q$, and we show related associativity and commutativity results. This gives rise to the notion of the \"orbit of a box\", unveiling surprising geometrical properties about the alignment and parallelism of distilled boxes. The power of this new framework is that it allows us to prove previously-reported numerical observations concerning the best way to wire consecutive boxes, and to numerically and analytically recover recently-identified noisy $texttt{PR}$ boxes that collapse communication complexity for different types of noise models.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566267","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 : 2024-07-10DOI: 10.22331/q-2024-07-10-1405
J. Escandón-Monardes, D. Uzcátegui, M. Rivera-Tapia, S. P. Walborn, A. Delgado
We propose an estimation procedure for $d$-dimensional unitary transformations. For $dgt2$, the unitary transformations close to the identity are estimated saturating the quantum Cramér-Rao bound. For $d=2$, the estimation of all unitary transformations is also optimal with some prior information. We show through numerical simulations that, even in the absence of prior information, two-dimensional unitary transformations can be estimated with greater precision than by means of standard quantum process tomography.
{"title":"Estimation of high-dimensional unitary transformations saturating the Quantum Cramér-Rao bound","authors":"J. Escandón-Monardes, D. Uzcátegui, M. Rivera-Tapia, S. P. Walborn, A. Delgado","doi":"10.22331/q-2024-07-10-1405","DOIUrl":"https://doi.org/10.22331/q-2024-07-10-1405","url":null,"abstract":"We propose an estimation procedure for $d$-dimensional unitary transformations. For $dgt2$, the unitary transformations close to the identity are estimated saturating the quantum Cramér-Rao bound. For $d=2$, the estimation of all unitary transformations is also optimal with some prior information. We show through numerical simulations that, even in the absence of prior information, two-dimensional unitary transformations can be estimated with greater precision than by means of standard quantum process tomography.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584496","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 : 2024-07-10DOI: 10.22331/q-2024-07-10-1401
Kshiti Sneh Rai, J. Ignacio Cirac, Álvaro M. Alhambra
We show how to efficiently simulate pure quantum states in one dimensional systems that have both finite energy density and vanishingly small energy fluctuations. We do so by studying the performance of a tensor network algorithm that produces matrix product states whose energy variance decreases as the bond dimension increases. Our results imply that variances as small as $propto 1/log N$ can be achieved with polynomial bond dimension. With this, we prove that there exist states with a very narrow support in the bulk of the spectrum that still have moderate entanglement entropy, in contrast with typical eigenstates that display a volume law. Our main technical tool is the Berry-Esseen theorem for spin systems, a strengthening of the central limit theorem for the energy distribution of product states. We also give a simpler proof of that theorem, together with slight improvements in the error scaling, which should be of independent interest.
{"title":"Matrix product state approximations to quantum states of low energy variance","authors":"Kshiti Sneh Rai, J. Ignacio Cirac, Álvaro M. Alhambra","doi":"10.22331/q-2024-07-10-1401","DOIUrl":"https://doi.org/10.22331/q-2024-07-10-1401","url":null,"abstract":"We show how to efficiently simulate pure quantum states in one dimensional systems that have both finite energy density and vanishingly small energy fluctuations. We do so by studying the performance of a tensor network algorithm that produces matrix product states whose energy variance decreases as the bond dimension increases. Our results imply that variances as small as $propto 1/log N$ can be achieved with polynomial bond dimension. With this, we prove that there exist states with a very narrow support in the bulk of the spectrum that still have moderate entanglement entropy, in contrast with typical eigenstates that display a volume law. Our main technical tool is the Berry-Esseen theorem for spin systems, a strengthening of the central limit theorem for the energy distribution of product states. We also give a simpler proof of that theorem, together with slight improvements in the error scaling, which should be of independent interest.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":6.4,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566266","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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