Pub Date : 2024-11-21DOI: 10.22331/q-2024-11-21-1533
D. Fernández-Fernández, Yue Ban, G. Platero
Quantum information transfer is fundamental for scalable quantum computing in any potential platform and architecture. Hole spin qubits, owing to their intrinsic spin-orbit interaction (SOI), promise fast quantum operations which are fundamental for the implementation of quantum gates. Yet, the influence of SOI in quantum transfer protocols remains an open question. Here, we investigate flying spin qubits mediated by SOI, using shortcuts to adiabaticity protocols, i.e., the long-range transfer of spin qubits and the quantum distribution of entangled pairs in semiconductor quantum dot arrays. We show that electric field manipulation allows dynamical control of the SOI, enabling simultaneously the implementation of quantum gates during the transfer, with the potential to significantly accelerate quantum algorithms. By harnessing the ability to perform quantum gates in parallel with the transfer, we implement dynamical decoupling schemes to focus and preserve the spin state, leading to higher transfer fidelity.
量子信息传输是任何潜在平台和架构中可扩展量子计算的基础。空穴自旋量子比特因其固有的自旋轨道相互作用（SOI）而有望实现快速量子操作，这对量子门的实现至关重要。然而，SOI 对量子传输协议的影响仍是一个未决问题。在这里，我们利用绝热协议的捷径，即自旋量子比特的长程传输和纠缠对在半导体量子点阵列中的量子分布，研究了由 SOI 介导的飞行自旋量子比特。我们的研究表明，电场操纵可实现对 SOI 的动态控制，在传输过程中同时实现量子门，从而有可能显著加速量子算法。通过利用量子门与转移并行执行的能力，我们实施了动态解耦方案来聚焦和保留自旋状态，从而提高了转移的保真度。
{"title":"Flying Spin Qubits in Quantum Dot Arrays Driven by Spin-Orbit Interaction","authors":"D. Fernández-Fernández, Yue Ban, G. Platero","doi":"10.22331/q-2024-11-21-1533","DOIUrl":"https://doi.org/10.22331/q-2024-11-21-1533","url":null,"abstract":"Quantum information transfer is fundamental for scalable quantum computing in any potential platform and architecture. Hole spin qubits, owing to their intrinsic spin-orbit interaction (SOI), promise fast quantum operations which are fundamental for the implementation of quantum gates. Yet, the influence of SOI in quantum transfer protocols remains an open question. Here, we investigate flying spin qubits mediated by SOI, using shortcuts to adiabaticity protocols, i.e., the long-range transfer of spin qubits and the quantum distribution of entangled pairs in semiconductor quantum dot arrays. We show that electric field manipulation allows dynamical control of the SOI, enabling simultaneously the implementation of quantum gates during the transfer, with the potential to significantly accelerate quantum algorithms. By harnessing the ability to perform quantum gates in parallel with the transfer, we implement dynamical decoupling schemes to focus and preserve the spin state, leading to higher transfer fidelity.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"25 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678474","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 : 2024-11-21DOI: 10.22331/q-2024-11-21-1534
Giovanni Di Meglio, Martin B. Plenio, Susana F. Huelga
We derive a Markovian master equation that models the evolution of systems subject to driving and control fields. Our approach combines time rescaling and weak-coupling limits for the system-environment interaction with a secular approximation. The derivation makes use of the adiabatic time-evolution operator in a manner that allows for the efficient description of strong driving, while recovering the well-known adiabatic master equation in the appropriate limit. To illustrate the effectiveness of our approach, firstly we apply it to the paradigmatic case of a two-level (qubit) system subject to a form of periodic driving that remains unsolvable using a Floquet representation and lastly we extend this scenario to the situation of two interacting qubits, the first driven while the second one directly in contact with the environment. We demonstrate the reliability and broad scope of our approach by benchmarking the solutions of the derived reduced time evolution against numerically exact simulations using tensor networks. Our results provide rigorous conditions that must be satisfied by phenomenological master equations for driven systems that do not rely on first-principles derivations.
{"title":"Time dependent Markovian master equation beyond the adiabatic limit","authors":"Giovanni Di Meglio, Martin B. Plenio, Susana F. Huelga","doi":"10.22331/q-2024-11-21-1534","DOIUrl":"https://doi.org/10.22331/q-2024-11-21-1534","url":null,"abstract":"We derive a Markovian master equation that models the evolution of systems subject to driving and control fields. Our approach combines time rescaling and weak-coupling limits for the system-environment interaction with a secular approximation. The derivation makes use of the adiabatic time-evolution operator in a manner that allows for the efficient description of strong driving, while recovering the well-known adiabatic master equation in the appropriate limit. To illustrate the effectiveness of our approach, firstly we apply it to the paradigmatic case of a two-level (qubit) system subject to a form of periodic driving that remains unsolvable using a Floquet representation and lastly we extend this scenario to the situation of two interacting qubits, the first driven while the second one directly in contact with the environment. We demonstrate the reliability and broad scope of our approach by benchmarking the solutions of the derived reduced time evolution against numerically exact simulations using tensor networks. Our results provide rigorous conditions that must be satisfied by phenomenological master equations for driven systems that do not rely on first-principles derivations.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"11 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678476","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 : 2024-11-20DOI: 10.22331/q-2024-11-20-1532
Jesse Berwald, Nicholas Chancellor, Raouf Dridi
It has previously been established that adiabatic quantum computation, operating based on a continuous Zeno effect due to dynamical phases between eigenstates, is able to realise an optimal Grover-like quantum speedup. In other words, is able to solve an unstructured search problem with the same $sqrt{N}$ scaling as Grover's original algorithm. A natural question is whether other manifestations of the Zeno effect can also support an optimal speedup in a physically realistic model (through direct analogue application rather than indirectly by supporting a universal gateset). In this paper we show that they can support such a speedup, whether due to measurement, decoherence, or even decay of the excited state into a computationally useless state. Our results also suggest a wide variety of methods to realise speedup which do not rely on Zeno behaviour. We group these algorithms into three families to facilitate a structured understanding of how speedups can be obtained: one based on phase kicks, containing adiabatic computation and continuous-time quantum walks; one based on dephasing and measurement; and finally one based on destruction of the amplitude within the excited state, for which we are not aware of any previous results. These results suggest that there may be exciting opportunities for new paradigms of analog quantum computing based on these effects.
{"title":"Grover Speedup from Many Forms of the Zeno Effect","authors":"Jesse Berwald, Nicholas Chancellor, Raouf Dridi","doi":"10.22331/q-2024-11-20-1532","DOIUrl":"https://doi.org/10.22331/q-2024-11-20-1532","url":null,"abstract":"It has previously been established that adiabatic quantum computation, operating based on a continuous Zeno effect due to dynamical phases between eigenstates, is able to realise an optimal Grover-like quantum speedup. In other words, is able to solve an unstructured search problem with the same $sqrt{N}$ scaling as Grover's original algorithm. A natural question is whether other manifestations of the Zeno effect can also support an optimal speedup in a physically realistic model (through direct analogue application rather than indirectly by supporting a universal gateset). In this paper we show that they can support such a speedup, whether due to measurement, decoherence, or even decay of the excited state into a computationally useless state. Our results also suggest a wide variety of methods to realise speedup which do not rely on Zeno behaviour. We group these algorithms into three families to facilitate a structured understanding of how speedups can be obtained: one based on phase kicks, containing adiabatic computation and continuous-time quantum walks; one based on dephasing and measurement; and finally one based on destruction of the amplitude within the excited state, for which we are not aware of any previous results. These results suggest that there may be exciting opportunities for new paradigms of analog quantum computing based on these effects.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"35 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678472","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 : 2024-11-20DOI: 10.22331/q-2024-11-20-1528
Suhail Ahmad Rather
Dual unitary gates are highly non-local two-qudit unitary gates that have been studied extensively in quantum many-body physics and quantum information in the recent past. A special class of dual unitary gates consists of rank-four perfect tensors that are equivalent to highly entangled multipartite pure states called absolutely maximally entangled (AME) states. In this work, numerical and analytical constructions of dual unitary gates and perfect tensors that are diagonal in a special maximally entangled basis are presented. The main ingredient in our construction is a phase-valued (unimodular) two-dimensional array whose discrete Fourier transform is also unimodular. We obtain perfect tensors for several local Hilbert space dimensions, particularly, in dimension six. A perfect tensor in local dimension six is equivalent to an AME state of four qudits, denoted as AME(4,6). Such a state cannot be constructed from existing constructions of AME states based on error-correcting codes and graph states. An explicit construction of AME(4,6) states is provided in this work using two-qudit controlled and single-qudit gates making it feasible to generate such states experimentally.
{"title":"Construction of perfect tensors using biunimodular vectors","authors":"Suhail Ahmad Rather","doi":"10.22331/q-2024-11-20-1528","DOIUrl":"https://doi.org/10.22331/q-2024-11-20-1528","url":null,"abstract":"Dual unitary gates are highly non-local two-qudit unitary gates that have been studied extensively in quantum many-body physics and quantum information in the recent past. A special class of dual unitary gates consists of rank-four perfect tensors that are equivalent to highly entangled multipartite pure states called absolutely maximally entangled (AME) states. In this work, numerical and analytical constructions of dual unitary gates and perfect tensors that are diagonal in a special maximally entangled basis are presented. The main ingredient in our construction is a phase-valued (unimodular) two-dimensional array whose discrete Fourier transform is also unimodular. We obtain perfect tensors for several local Hilbert space dimensions, particularly, in dimension six. A perfect tensor in local dimension six is equivalent to an AME state of four qudits, denoted as AME(4,6). Such a state cannot be constructed from existing constructions of AME states based on error-correcting codes and graph states. An explicit construction of AME(4,6) states is provided in this work using two-qudit controlled and single-qudit gates making it feasible to generate such states experimentally.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"14 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142673918","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 : 2024-11-20DOI: 10.22331/q-2024-11-20-1529
Gilad Gour, Mark M. Wilde, S. Brandsen, Isabelle Jianing Geng
A colloquial interpretation of entropy is that it is the knowledge gained upon learning the outcome of a random experiment. Conditional entropy is then interpreted as the knowledge gained upon learning the outcome of one random experiment after learning the outcome of another, possibly statistically dependent, random experiment. In the classical world, entropy and conditional entropy take only non-negative values, consistent with the intuition that one has regarding the aforementioned interpretations. However, for certain entangled states, one obtains negative values when evaluating commonly accepted and information-theoretically justified formulas for the quantum conditional entropy, leading to the confounding conclusion that one can know less than nothing in the quantum world. Here, we introduce a physically motivated framework for defining quantum conditional entropy, based on two simple postulates inspired by the second law of thermodynamics (non-decrease of entropy) and extensivity of entropy, and we argue that all plausible definitions of quantum conditional entropy should respect these two postulates. We then prove that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world. All of our arguments are based on constructions of physical processes that respect the first postulate, the one inspired by the second law of thermodynamics.
{"title":"Inevitability of knowing less than nothing","authors":"Gilad Gour, Mark M. Wilde, S. Brandsen, Isabelle Jianing Geng","doi":"10.22331/q-2024-11-20-1529","DOIUrl":"https://doi.org/10.22331/q-2024-11-20-1529","url":null,"abstract":"A colloquial interpretation of entropy is that it is the knowledge gained upon learning the outcome of a random experiment. Conditional entropy is then interpreted as the knowledge gained upon learning the outcome of one random experiment after learning the outcome of another, possibly statistically dependent, random experiment. In the classical world, entropy and conditional entropy take only non-negative values, consistent with the intuition that one has regarding the aforementioned interpretations. However, for certain entangled states, one obtains negative values when evaluating commonly accepted and information-theoretically justified formulas for the quantum conditional entropy, leading to the confounding conclusion that one can know less than nothing in the quantum world. Here, we introduce a physically motivated framework for defining quantum conditional entropy, based on two simple postulates inspired by the second law of thermodynamics (non-decrease of entropy) and extensivity of entropy, and we argue that all plausible definitions of quantum conditional entropy should respect these two postulates. We then prove that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world. All of our arguments are based on constructions of physical processes that respect the first postulate, the one inspired by the second law of thermodynamics.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"99 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142673919","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 : 2024-11-20DOI: 10.22331/q-2024-11-20-1530
Jonathan Allcock, Jinge Bao, Joao F. Doriguello, Alessandro Luongo, Miklos Santha
We explore the power of the unbounded Fan-Out gate and the Global Tunable gates generated by Ising-type Hamiltonians in constructing constant-depth quantum circuits, with particular attention to quantum memory devices. We propose two types of constant-depth constructions for implementing Uniformly Controlled Gates. These gates include the Fan-In gates defined by $|xrangle|branglemapsto |xrangle|boplus f(x)rangle$ for $xin{0,1}^n$ and $bin{0,1}$, where $f$ is a Boolean function. The first of our constructions is based on computing the one-hot encoding of the control register $|xrangle$, while the second is based on Boolean analysis and exploits different representations of $f$ such as its Fourier expansion. Via these constructions, we obtain constant-depth circuits for the quantum counterparts of read-only and read-write memory devices – Quantum Random Access Memory (QRAM) and Quantum Random Access Gate (QRAG) – of memory size $n$. The implementation based on one-hot encoding requires either $O(nlog^{(d)}{n}log^{(d+1)}{n})$ ancillae and $O(nlog^{(d)}{n})$ Fan-Out gates or $O(nlog^{(d)}{n})$ ancillae and $16d-10$ Global Tunable gates, where $d$ is any positive integer and $log^{(d)}{n} = logcdots log{n}$ is the $d$-times iterated logarithm. On the other hand, the implementation based on Boolean analysis requires $8d-6$ Global Tunable gates at the expense of $O(n^{1/(1-2^{-d})})$ ancillae.
{"title":"Constant-depth circuits for Boolean functions and quantum memory devices using multi-qubit gates","authors":"Jonathan Allcock, Jinge Bao, Joao F. Doriguello, Alessandro Luongo, Miklos Santha","doi":"10.22331/q-2024-11-20-1530","DOIUrl":"https://doi.org/10.22331/q-2024-11-20-1530","url":null,"abstract":"We explore the power of the unbounded Fan-Out gate and the Global Tunable gates generated by Ising-type Hamiltonians in constructing constant-depth quantum circuits, with particular attention to quantum memory devices. We propose two types of constant-depth constructions for implementing Uniformly Controlled Gates. These gates include the Fan-In gates defined by $|xrangle|branglemapsto |xrangle|boplus f(x)rangle$ for $xin{0,1}^n$ and $bin{0,1}$, where $f$ is a Boolean function. The first of our constructions is based on computing the one-hot encoding of the control register $|xrangle$, while the second is based on Boolean analysis and exploits different representations of $f$ such as its Fourier expansion. Via these constructions, we obtain constant-depth circuits for the quantum counterparts of read-only and read-write memory devices – Quantum Random Access Memory (QRAM) and Quantum Random Access Gate (QRAG) – of memory size $n$. The implementation based on one-hot encoding requires either $O(nlog^{(d)}{n}log^{(d+1)}{n})$ ancillae and $O(nlog^{(d)}{n})$ Fan-Out gates or $O(nlog^{(d)}{n})$ ancillae and $16d-10$ Global Tunable gates, where $d$ is any positive integer and $log^{(d)}{n} = logcdots log{n}$ is the $d$-times iterated logarithm. On the other hand, the implementation based on Boolean analysis requires $8d-6$ Global Tunable gates at the expense of $O(n^{1/(1-2^{-d})})$ ancillae.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"35 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142673920","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 : 2024-11-20DOI: 10.22331/q-2024-11-20-1531
Rutuja Kshirsagar, Amara Katabarwa, Peter D. Johnson
The hope of the quantum computing field is that quantum architectures are able to scale up and realize fault-tolerant quantum computing. Due to engineering challenges, such ''cheap'' error correction may be decades away. In the meantime, we anticipate an era of ''costly'' error correction, or $textit{early fault-tolerant quantum computing}$. Costly error correction might warrant settling for error-prone quantum computations. This motivates the development of quantum algorithms which are robust to some degree of error as well as methods to analyze their performance in the presence of error. Several such algorithms have recently been developed; what is missing is a methodology to analyze their robustness. To this end, we introduce a randomized algorithm for the task of phase estimation and give an analysis of its performance under two simple noise models. In both cases the analysis leads to a noise threshold, below which arbitrarily high accuracy can be achieved by increasing the number of samples used in the algorithm. As an application of this general analysis, we compute the maximum ratio of the largest circuit depth and the dephasing scale such that performance guarantees hold. We calculate that the randomized algorithm can succeed with arbitrarily high probability as long as the required circuit depth is less than 0.916 times the dephasing scale.
{"title":"On proving the robustness of algorithms for early fault-tolerant quantum computers","authors":"Rutuja Kshirsagar, Amara Katabarwa, Peter D. Johnson","doi":"10.22331/q-2024-11-20-1531","DOIUrl":"https://doi.org/10.22331/q-2024-11-20-1531","url":null,"abstract":"The hope of the quantum computing field is that quantum architectures are able to scale up and realize fault-tolerant quantum computing. Due to engineering challenges, such ''cheap'' error correction may be decades away. In the meantime, we anticipate an era of ''costly'' error correction, or $textit{early fault-tolerant quantum computing}$. Costly error correction might warrant settling for error-prone quantum computations. This motivates the development of quantum algorithms which are robust to some degree of error as well as methods to analyze their performance in the presence of error. Several such algorithms have recently been developed; what is missing is a methodology to analyze their robustness. To this end, we introduce a randomized algorithm for the task of phase estimation and give an analysis of its performance under two simple noise models. In both cases the analysis leads to a noise threshold, below which arbitrarily high accuracy can be achieved by increasing the number of samples used in the algorithm. As an application of this general analysis, we compute the maximum ratio of the largest circuit depth and the dephasing scale such that performance guarantees hold. We calculate that the randomized algorithm can succeed with arbitrarily high probability as long as the required circuit depth is less than 0.916 times the dephasing scale.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"6 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678473","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 : 2024-11-19DOI: 10.22331/q-2024-11-19-1527
Eric R. Anschuetz, David Gamarnik, Bobak Kiani
In an important recent development, Anshu, Breuckmann, and Nirkhe [3] resolved positively the so-called No Low-Energy Trivial State (NLTS) conjecture by Freedman and Hastings. The conjecture postulated the existence of linear-size local Hamiltonians on n qubit systems for which no near-ground state can be prepared by a shallow (sublogarithmic depth) circuit. The construction in [3] is based on recently developed good quantum codes. Earlier results in this direction included the constructions of the so-called Combinatorial NLTS – a weaker version of NLTS – where a state is defined to have low energy if it violates at most a vanishing fraction of the Hamiltonian terms [2]. These constructions were also based on codes. In this paper we provide a "non-code" construction of a class of Hamiltonians satisfying the Combinatorial NLTS. The construction is inspired by one in [2], but our proof uses the complex solution space geometry of random K-SAT instead of properties of codes. Specifically, it is known that above a certain clause-to-variables density the set of satisfying assignments of random K-SAT exhibits an overlap gap property, which implies that it can be partitioned into exponentially many clusters each constituting at most an exponentially small fraction of the total set of satisfying solutions. We establish a certain robust version of this clustering property for the space of near-satisfying assignments and show that for our constructed Hamiltonians every combinatorial near-ground state induces a near-uniform distribution supported by this set. Standard arguments then are used to show that such distributions cannot be prepared by quantum circuits with depth o(log n). Since the clustering property is exhibited by many random structures, including proper coloring and maximum cut, we anticipate that our approach is extendable to these models as well.
{"title":"Combinatorial NLTS From the Overlap Gap Property","authors":"Eric R. Anschuetz, David Gamarnik, Bobak Kiani","doi":"10.22331/q-2024-11-19-1527","DOIUrl":"https://doi.org/10.22331/q-2024-11-19-1527","url":null,"abstract":"In an important recent development, Anshu, Breuckmann, and Nirkhe [3] resolved positively the so-called No Low-Energy Trivial State (NLTS) conjecture by Freedman and Hastings. The conjecture postulated the existence of linear-size local Hamiltonians on n qubit systems for which no near-ground state can be prepared by a shallow (sublogarithmic depth) circuit. The construction in [3] is based on recently developed good quantum codes. Earlier results in this direction included the constructions of the so-called Combinatorial NLTS – a weaker version of NLTS – where a state is defined to have low energy if it violates at most a vanishing fraction of the Hamiltonian terms [2]. These constructions were also based on codes.<br/> In this paper we provide a \"non-code\" construction of a class of Hamiltonians satisfying the Combinatorial NLTS. The construction is inspired by one in [2], but our proof uses the complex solution space geometry of random K-SAT instead of properties of codes. Specifically, it is known that above a certain clause-to-variables density the set of satisfying assignments of random K-SAT exhibits an overlap gap property, which implies that it can be partitioned into exponentially many clusters each constituting at most an exponentially small fraction of the total set of satisfying solutions. We establish a certain robust version of this clustering property for the space of near-satisfying assignments and show that for our constructed Hamiltonians every combinatorial near-ground state induces a near-uniform distribution supported by this set. Standard arguments then are used to show that such distributions cannot be prepared by quantum circuits with depth o(log n). Since the clustering property is exhibited by many random structures, including proper coloring and maximum cut, we anticipate that our approach is extendable to these models as well.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"69 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142670912","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 : 2024-11-18DOI: 10.22331/q-2024-11-18-1526
Jop Briët, Francisco Escudero Gutiérrez, Sander Gribling
A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.
{"title":"Grothendieck inequalities characterize converses to the polynomial method","authors":"Jop Briët, Francisco Escudero Gutiérrez, Sander Gribling","doi":"10.22331/q-2024-11-18-1526","DOIUrl":"https://doi.org/10.22331/q-2024-11-18-1526","url":null,"abstract":"A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"64 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142670910","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 : 2024-11-14DOI: 10.22331/q-2024-11-14-1525
Lexin Ding, Cheng-Lin Hong, Christian Schilling
The extension of the Rayleigh-Ritz variational principle to ensemble states $rho_{mathbf{w}}equivsum_k w_k |Psi_krangle langlePsi_k|$ with fixed weights $w_k$ lies ultimately at the heart of several recent methodological developments for targeting excitation energies by variational means. Prominent examples are density and density matrix functional theory, Monte Carlo sampling, state-average complete active space self-consistent field methods and variational quantum eigensolvers. In order to provide a sound basis for all these methods and to improve their current implementations, we prove the validity of the underlying critical hypothesis: Whenever the ensemble energy is well-converged, the same holds true for the ensemble state $rho_{mathbf{w}}$ as well as the individual eigenstates $|Psi_krangle$ and eigenenergies $E_k$. To be more specific, we derive linear bounds $d_-Delta{E}_{mathbf{w}} leq Delta Q leq d_+ Delta{E}_{mathbf{w}}$ on the errors $Delta Q $ of these sought-after quantities. A subsequent analytical analysis and numerical illustration proves the tightness of our universal inequalities. Our results and particularly the explicit form of $d_{pm}equiv d_{pm}^{(Q)}(mathbf{w},mathbf{E})$ provide valuable insights into the optimal choice of the auxiliary weights $w_k$ in practical applications.
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