Pub Date : 2024-10-30DOI: 10.22331/q-2024-10-30-1514
Hippolyte Dourdent, Alastair A. Abbott, Ivan Šupić, Cyril Branciard
Causal nonseparability is the property underlying quantum processes incompatible with a definite causal order. So far it has remained a central open question as to whether any process with a clear physical realisation can violate a causal inequality, so that its causal nonseparability can be certified in a device-independent way, as originally conceived. Here we present a method solely based on the observed correlations, which certifies the causal nonseparability of all the processes that can induce a causally nonseparable distributed measurement in a scenario with trusted quantum input states, as defined in [Dourdent et al., Phys. Rev. Lett. 129, 090402 (2022)]. This notably includes the celebrated quantum switch. This device-independent certification is achieved by introducing a network of untrusted operations, allowing one to self-test the quantum inputs on which the effective distributed measurement induced by the process is performed.
{"title":"Network-Device-Independent Certification of Causal Nonseparability","authors":"Hippolyte Dourdent, Alastair A. Abbott, Ivan Šupić, Cyril Branciard","doi":"10.22331/q-2024-10-30-1514","DOIUrl":"https://doi.org/10.22331/q-2024-10-30-1514","url":null,"abstract":"Causal nonseparability is the property underlying quantum processes incompatible with a definite causal order. So far it has remained a central open question as to whether any process with a clear physical realisation can violate a causal inequality, so that its causal nonseparability can be certified in a device-independent way, as originally conceived. Here we present a method solely based on the observed correlations, which certifies the causal nonseparability of all the processes that can induce a causally nonseparable distributed measurement in a scenario with trusted quantum input states, as defined in [Dourdent et al., Phys. Rev. Lett. 129, 090402 (2022)]. This notably includes the celebrated quantum switch. This device-independent certification is achieved by introducing a network of untrusted operations, allowing one to self-test the quantum inputs on which the effective distributed measurement induced by the process is performed.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"38 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142556033","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 : 2024-10-29DOI: 10.22331/q-2024-10-29-1512
Hitomi Mori, Kaoru Mizuta, Keisuke Fujii
Multivariable Quantum Signal Processing (M-QSP) [1] is expected to provide an efficient means to handle polynomial transformations of multiple variables simultaneously. However, we identified several inconsistencies in the main theorem, where necessary and sufficient conditions for achievable polynomials are provided, and its proof in Ref. [1]. Moreover, a counterexample to the conjecture in Ref. [1], based on which the main theorem is constructed, is presented in Ref. [2], meaning the requirement of the conjecture should be included as a condition in the theorem. Here we note our observations and propose the revised necessary conditions for M-QSP. We also show that these necessary conditions cannot be sufficient conditions, and thus some additional condition on top of these revisions is essentially required for the complete M-QSP theorem.
{"title":"Comment on “Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle”","authors":"Hitomi Mori, Kaoru Mizuta, Keisuke Fujii","doi":"10.22331/q-2024-10-29-1512","DOIUrl":"https://doi.org/10.22331/q-2024-10-29-1512","url":null,"abstract":"Multivariable Quantum Signal Processing (M-QSP) [1] is expected to provide an efficient means to handle polynomial transformations of multiple variables simultaneously. However, we identified several inconsistencies in the main theorem, where necessary and sufficient conditions for achievable polynomials are provided, and its proof in Ref. [1]. Moreover, a counterexample to the conjecture in Ref. [1], based on which the main theorem is constructed, is presented in Ref. [2], meaning the requirement of the conjecture should be included as a condition in the theorem. Here we note our observations and propose the revised necessary conditions for M-QSP. We also show that these necessary conditions cannot be sufficient conditions, and thus some additional condition on top of these revisions is essentially required for the complete M-QSP theorem.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"25 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142536699","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 : 2024-10-29DOI: 10.22331/q-2024-10-29-1513
Matteo Votto, Johannes Zeiher, Benoît Vermersch
We propose a protocol to realize quantum simulation and computation in spin systems with long-range interactions. Our approach relies on the local addressing of single spins with external fields parametrized by Walsh functions. This enables a mapping from a class of target Hamiltonians, defined by the graph structure of their interactions, to pulse sequences. We then obtain a recipe to implement arbitrary two-body Hamiltonians and universal quantum circuits. Performance guarantees are provided in terms of bounds on Trotter errors and total number of pulses. Additionally, Walsh pulse sequences are shown to be robust against various types of pulse errors, in contrast to previous hybrid digital-analog schemes of quantum computation. We demonstrate and numerically benchmark our protocol with examples from the dynamics of spin models, quantum error correction and quantum optimization algorithms.
{"title":"Universal quantum processors in spin systems via robust local pulse sequences","authors":"Matteo Votto, Johannes Zeiher, Benoît Vermersch","doi":"10.22331/q-2024-10-29-1513","DOIUrl":"https://doi.org/10.22331/q-2024-10-29-1513","url":null,"abstract":"We propose a protocol to realize quantum simulation and computation in spin systems with long-range interactions. Our approach relies on the local addressing of single spins with external fields parametrized by Walsh functions. This enables a mapping from a class of target Hamiltonians, defined by the graph structure of their interactions, to pulse sequences. We then obtain a recipe to implement arbitrary two-body Hamiltonians and universal quantum circuits. Performance guarantees are provided in terms of bounds on Trotter errors and total number of pulses. Additionally, Walsh pulse sequences are shown to be robust against various types of pulse errors, in contrast to previous hybrid digital-analog schemes of quantum computation. We demonstrate and numerically benchmark our protocol with examples from the dynamics of spin models, quantum error correction and quantum optimization algorithms.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"100 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142536700","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 : 2024-10-28DOI: 10.22331/q-2024-10-28-1511
Alex Fischer, Akimasa Miyake
Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit.
In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent $X$ and $Z$ noise.
{"title":"Hardness results for decoding the surface code with Pauli noise","authors":"Alex Fischer, Akimasa Miyake","doi":"10.22331/q-2024-10-28-1511","DOIUrl":"https://doi.org/10.22331/q-2024-10-28-1511","url":null,"abstract":"Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit.<br/>\u0000<br/> In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent $X$ and $Z$ noise.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"15 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142536698","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 : 2024-10-24DOI: 10.22331/q-2024-10-24-1510
David Raveh, Rafael I. Nepomechie
We present an explicit quantum circuit that prepares an arbitrary $U(1)$-eigenstate on a quantum computer, including the exact eigenstates of the spin-$1/2 XXZ$ quantum spin chain with either open or closed boundary conditions. The algorithm is deterministic, does not require ancillary qubits, and does not require QR decompositions. The circuit prepares such an $L$-qubit state with $M$ down-spins using $binom{L}{M}-1$ multi-controlled rotation gates and $2M(L-M)$ CNOT-gates.
{"title":"Deterministic Bethe state preparation","authors":"David Raveh, Rafael I. Nepomechie","doi":"10.22331/q-2024-10-24-1510","DOIUrl":"https://doi.org/10.22331/q-2024-10-24-1510","url":null,"abstract":"We present an explicit quantum circuit that prepares an arbitrary $U(1)$-eigenstate on a quantum computer, including the exact eigenstates of the spin-$1/2 XXZ$ quantum spin chain with either open or closed boundary conditions. The algorithm is deterministic, does not require ancillary qubits, and does not require QR decompositions. The circuit prepares such an $L$-qubit state with $M$ down-spins using $binom{L}{M}-1$ multi-controlled rotation gates and $2M(L-M)$ CNOT-gates.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"25 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142488932","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 : 2024-10-24DOI: 10.22331/q-2024-10-24-1509
Dylan Lewis, Stephan Eidenbenz, Balasubramanya Nadiga, Yiğit Subaşı
We investigate the limitations of quantum computers for solving nonlinear dynamical systems. In particular, we tighten the worst-case bounds of the quantum Carleman linearisation (QCL) algorithm [Liu et al., PNAS 118, 2021] answering one of their open questions. We provide a further significant limitation for any quantum algorithm that aims to output a quantum state that approximates the normalized solution vector. Given a natural choice of coordinates for a dynamical system with one or more positive Lyapunov exponents and solutions that grow sub-exponentially, we prove that any such algorithm has complexity scaling at least exponentially in the integration time. As such, an efficient quantum algorithm for simulating chaotic systems or regimes is likely not possible.
{"title":"Limitations for Quantum Algorithms to Solve Turbulent and Chaotic Systems","authors":"Dylan Lewis, Stephan Eidenbenz, Balasubramanya Nadiga, Yiğit Subaşı","doi":"10.22331/q-2024-10-24-1509","DOIUrl":"https://doi.org/10.22331/q-2024-10-24-1509","url":null,"abstract":"We investigate the limitations of quantum computers for solving nonlinear dynamical systems. In particular, we tighten the worst-case bounds of the quantum Carleman linearisation (QCL) algorithm [Liu et al., PNAS 118, 2021] answering one of their open questions. We provide a further significant limitation for any quantum algorithm that aims to output a quantum state that approximates the normalized solution vector. Given a natural choice of coordinates for a dynamical system with one or more positive Lyapunov exponents and solutions that grow sub-exponentially, we prove that any such algorithm has complexity scaling at least exponentially in the integration time. As such, an efficient quantum algorithm for simulating chaotic systems or regimes is likely not possible.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"3 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142488989","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 : 2024-10-23DOI: 10.22331/q-2024-10-23-1508
Pablo Arrighi, Amélia Durbec, Matt Wilson
The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of being connected or not, and be allowed to merge, split and reconnect coherently in a superposition. Second, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates in a robust manner. The hereby presented mathematical framework is anchored on solid grounds through numerous lemmas. Indeed, one might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeopardized as the neighbourhood and partitioning between systems become both quantum, dynamical, and logical. Such interrelations in fact carry through, albeit two new notions become instrumental: consistency and comprehension.
{"title":"Quantum networks theory","authors":"Pablo Arrighi, Amélia Durbec, Matt Wilson","doi":"10.22331/q-2024-10-23-1508","DOIUrl":"https://doi.org/10.22331/q-2024-10-23-1508","url":null,"abstract":"The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of being connected or not, and be allowed to merge, split and reconnect coherently in a superposition. Second, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates in a robust manner. The hereby presented mathematical framework is anchored on solid grounds through numerous lemmas. Indeed, one might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeopardized as the neighbourhood and partitioning between systems become both quantum, dynamical, and logical. Such interrelations in fact carry through, albeit two new notions become instrumental: consistency and comprehension.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"234 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487368","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 : 2024-10-23DOI: 10.22331/q-2024-10-23-1505
Anne Matthies, Mark Rudner, Achim Rosch, Erez Berg
We propose a simple, robust protocol to prepare a low-energy state of an arbitrary Hamiltonian on a quantum computer or programmable quantum simulator. The protocol is inspired by the adiabatic demagnetization technique, used to cool solid-state systems to extremely low temperatures. A fraction of the qubits (or spins) is used to model a spin bath that is coupled to the system. By an adiabatic ramp down of a simulated Zeeman field acting on the bath spins, energy and entropy are extracted from the system. The bath spins are then measured and reset to the polarized state, and the process is repeated until convergence to a low-energy steady state is achieved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol's performance in the presence of noise and show how the information from the measurement of the bath spins can be used to monitor the cooling process. The performance of the algorithm depends on the nature of the excitations of the system; systems with non-local (topological) excitations are more difficult to cool than those with local excitations. We explore the possible mitigation of this problem by trapping topological excitations.
{"title":"Programmable adiabatic demagnetization for systems with trivial and topological excitations","authors":"Anne Matthies, Mark Rudner, Achim Rosch, Erez Berg","doi":"10.22331/q-2024-10-23-1505","DOIUrl":"https://doi.org/10.22331/q-2024-10-23-1505","url":null,"abstract":"We propose a simple, robust protocol to prepare a low-energy state of an arbitrary Hamiltonian on a quantum computer or programmable quantum simulator. The protocol is inspired by the adiabatic demagnetization technique, used to cool solid-state systems to extremely low temperatures. A fraction of the qubits (or spins) is used to model a spin bath that is coupled to the system. By an adiabatic ramp down of a simulated Zeeman field acting on the bath spins, energy and entropy are extracted from the system. The bath spins are then measured and reset to the polarized state, and the process is repeated until convergence to a low-energy steady state is achieved. We demonstrate the protocol via application to the quantum Ising model. We study the protocol's performance in the presence of noise and show how the information from the measurement of the bath spins can be used to monitor the cooling process. The performance of the algorithm depends on the nature of the excitations of the system; systems with non-local (topological) excitations are more difficult to cool than those with local excitations. We explore the possible mitigation of this problem by trapping topological excitations.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"1 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487365","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 : 2024-10-23DOI: 10.22331/q-2024-10-23-1504
Guoming Wang, Angus Kan
We develop quantum algorithms for pricing Asian and barrier options under the Heston model, a popular stochastic volatility model, and estimate their costs, in terms of T-count, T-depth and number of logical qubits, on instances under typical market conditions. These algorithms are based on combining well-established numerical methods for stochastic differential equations and quantum amplitude estimation technique. In particular, we empirically show that, despite its simplicity, weak Euler method achieves the same level of accuracy as the better-known strong Euler method in this task. Furthermore, by eliminating the expensive procedure of preparing Gaussian states, the quantum algorithm based on weak Euler scheme achieves drastically better efficiency than the one based on strong Euler scheme. Our resource analysis suggests that option pricing under stochastic volatility is a promising application of quantum computers, and that our algorithms render the hardware requirement for reaching practical quantum advantage in financial applications less stringent than prior art.
{"title":"Option pricing under stochastic volatility on a quantum computer","authors":"Guoming Wang, Angus Kan","doi":"10.22331/q-2024-10-23-1504","DOIUrl":"https://doi.org/10.22331/q-2024-10-23-1504","url":null,"abstract":"We develop quantum algorithms for pricing Asian and barrier options under the Heston model, a popular stochastic volatility model, and estimate their costs, in terms of T-count, T-depth and number of logical qubits, on instances under typical market conditions. These algorithms are based on combining well-established numerical methods for stochastic differential equations and quantum amplitude estimation technique. In particular, we empirically show that, despite its simplicity, weak Euler method achieves the same level of accuracy as the better-known strong Euler method in this task. Furthermore, by eliminating the expensive procedure of preparing Gaussian states, the quantum algorithm based on weak Euler scheme achieves drastically better efficiency than the one based on strong Euler scheme. Our resource analysis suggests that option pricing under stochastic volatility is a promising application of quantum computers, and that our algorithms render the hardware requirement for reaching practical quantum advantage in financial applications less stringent than prior art.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"1 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487582","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 : 2024-10-23DOI: 10.22331/q-2024-10-23-1507
Manuel B. Santos, Paulo Mateus, Chrysoula Vlachou
Oblivious linear evaluation is a generalization of oblivious transfer, whereby two distrustful parties obliviously compute a linear function, $f (x) = ax + b$, i.e., each one provides their inputs that remain unknown to the other, in order to compute the output $f (x)$ that only one of them receives. From both a structural and a security point of view, oblivious linear evaluation is fundamental for arithmetic-based secure multi-party computation protocols. In the classical case, oblivious linear evaluation protocols can be generated using oblivious transfer, and their quantum counterparts can, in principle, be constructed as straightforward extensions using quantum oblivious transfer. Here, we present the first, to the best of our knowledge, quantum protocol for oblivious linear evaluation that, furthermore, does not rely on quantum oblivious transfer. We start by presenting a semi-honest protocol, and then extend it to the dishonest setting employing a $commit-and-open$ strategy. Our protocol uses high-dimensional quantum states to obliviously compute $f (x)$ on Galois Fields of prime and prime-power dimension. These constructions utilize the existence of a complete set of mutually unbiased bases in prime-power dimension Hilbert spaces and their linear behaviour upon the Heisenberg-Weyl operators. We also generalize our protocol to achieve vector oblivious linear evaluation, where several instances of oblivious linear evaluation are generated, thus making the protocol more efficient. We prove the protocols to have static security in the framework of quantum universal composability.
{"title":"Quantum Universally Composable Oblivious Linear Evaluation","authors":"Manuel B. Santos, Paulo Mateus, Chrysoula Vlachou","doi":"10.22331/q-2024-10-23-1507","DOIUrl":"https://doi.org/10.22331/q-2024-10-23-1507","url":null,"abstract":"Oblivious linear evaluation is a generalization of oblivious transfer, whereby two distrustful parties obliviously compute a linear function, $f (x) = ax + b$, i.e., each one provides their inputs that remain unknown to the other, in order to compute the output $f (x)$ that only one of them receives. From both a structural and a security point of view, oblivious linear evaluation is fundamental for arithmetic-based secure multi-party computation protocols. In the classical case, oblivious linear evaluation protocols can be generated using oblivious transfer, and their quantum counterparts can, in principle, be constructed as straightforward extensions using quantum oblivious transfer. Here, we present the first, to the best of our knowledge, quantum protocol for oblivious linear evaluation that, furthermore, does not rely on quantum oblivious transfer. We start by presenting a semi-honest protocol, and then extend it to the dishonest setting employing a $commit-and-open$ strategy. Our protocol uses high-dimensional quantum states to obliviously compute $f (x)$ on Galois Fields of prime and prime-power dimension. These constructions utilize the existence of a complete set of mutually unbiased bases in prime-power dimension Hilbert spaces and their linear behaviour upon the Heisenberg-Weyl operators. We also generalize our protocol to achieve vector oblivious linear evaluation, where several instances of oblivious linear evaluation are generated, thus making the protocol more efficient. We prove the protocols to have static security in the framework of quantum universal composability.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":"2 1","pages":""},"PeriodicalIF":6.4,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487369","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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