Algorithms for Gibbs state preparation on noiseless and noisy random quantum circuits

Shtanko, Oles; Movassagh, Ramis

doi:10.48550/arxiv.2112.14688

Cited by 6 publications

(6 citation statements)

References 56 publications

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“…In one dimension, one can instead implement this same algorithm connecting subsequent segments of the chain, which can reduce the gate complexity to a polynomial ∼ N O(β) . Recent progress shows that the phase estimation and amplitude amplification steps in these schemes can instead be replaced by random circuits with post-selection [134], making them more amenable to current technologies. For commuting Hamiltonians, a purification in the form of a tensor network state (a PEPS) can be very efficiently prepared through an adiabatic algorithm [135].…”

Section: Algorithms Based On Purificationsmentioning

confidence: 99%

“…A later version of this algorithm is [141], in which the stochastic updates are accepted or not according to a quantum algorithm speeding up classical Markov processes [142]. A simpler algorithm, with a similar performance, can also be found in [134]. In all these, the convergence to the thermal distribution is guaranteed by the property of detailed balance.…”

Section: Quantum Metropolis Samplingmentioning

confidence: 99%

“…Some existing schemes do make use of relevant physical features to simplify them [43,132,135,143], but it seems that there is still plenty of room for exploring the kinds of regimes in which explicit and efficient algorithms can be proven. Since preparing thermal states is presumably an easier task than a general quantum computation (at least in certain regimes), it may be possible to tailor them to the limited capabilities of near-term noisy devices [134].…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

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Quantum many-body systems in thermal equilibrium

Alhambra¹

2022

Preprint

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The thermal or equilibrium ensemble is one of the most ubiquitous states of matter. For models comprised of many locally interacting quantum particles, it describes a wide range of physical situations, relevant to condensed matter physics, high energy physics, quantum chemistry and quantum computing, among others. We give a pedagogical overview of some of the most important universal features about the physics and complexity of these states, which have the locality of the Hamiltonian at its core. We focus on mathematically rigorous statements, many of them inspired by ideas and tools from quantum information theory. These include bounds on their correlations, the form of the subsystems, various statistical properties, and the performance of classical and quantum algorithms. We also include a summary of a few of the most important technical tools, as well as some self-contained proofs.Parts of these notes were the basis for a lecture series within the "Quantum Thermo-

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Section: Algorithms Based On Purificationsmentioning

confidence: 99%

Section: Quantum Metropolis Samplingmentioning

confidence: 99%

Section: Conclusion and Open Questionsmentioning

confidence: 99%

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Quantum many-body systems in thermal equilibrium

Alhambra¹

2022

Preprint

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“…Other quantum algorithms for thermal state preparation are also discussed in Ref. [58] where, under the ETH, the runtimes can be similar to that of QMS. (Note that the ETH does not always hold, as is the case for some integrable quantum systems [59].)…”

Section: Related Workmentioning

confidence: 99%

Quantum algorithms from fluctuation theorems: Thermal-state preparation

Holmes

Muraleedharan

Somma

et al. 2022

Quantum

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Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with Hamiltonians H0 and H1=H0+V. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H1 at inverse temperature β≥0 starting from a purification of the thermal state of H0. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is O~(eβ(ΔA−wl)/2), where ΔA is the free-energy difference between H1 and H0, and wl is a work cutoff that depends on the properties of the work distribution and the approximation error ϵ>0. If the non-equilibrium process is trivial, this complexity is exponential in β‖V‖, where ‖V‖ is the spectral norm of V. This represents a significant improvement of prior quantum algorithms that have complexity exponential in β‖H1‖ in the regime where ‖V‖≪‖H1‖. The dependence of the complexity in ϵ varies according to the structure of the quantum systems. It can be exponential in 1/ϵ in general, but we show it to be sublinear in 1/ϵ if H0 and H1 commute, or polynomial in 1/ϵ if H0 and H1 are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value of wl and improve the complexity even further. To this end, we analyze the complexity for preparing the thermal state of the transverse field Ising model using different non-equilibrium unitary processes and see significant complexity improvements.

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“…Strongly correlated physics such as Mottsuperfluid transition [11,12], unitary Fermi gas [13], and antiferromagnetism [14][15][16] have been accomplished with cold atoms. With rapid advancement of programmable quantum devices in the last several years such as superconducting qubits [17][18][19][20], trapped ions [21,22], entangled photons [23][24][25][26][27], and Rydberg atoms [28][29][30][31], there have been growing research interests in developing algorithmic approaches for quantum simulations [32][33][34][35][36][37]. Much progress has been made for determining ground states considering variants of quantum phase estimation [38,39], adiabatic Hamiltonian evolution [40,41], and variational quantum circuits [1,2,42,43].…”

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confidence: 99%

Quantum Kernel Function Expansion for Thermal Quantum Ensemble

Wang¹,

Jia²,

Zhang³

et al. 2022

Preprint

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Simulating quantum many-body systems is one major application of quantum computing, having a huge potential to impact the fields of computational physics and quantum chemistry. To fulfill the potential impacts, it is crucial to design quantum algorithms that efficiently employ the computation power of the quantum computing devices. Here, we introduce a quantum kernel function expansion algorithm for determining the thermodynamic quantities of quantum many-body systems, including local observables, free energy, and thermal entropy, which has been inspired by the kernel polynomial method in classical Hamiltonian simulation. We design quantum circuits to measure Fourier moments which constitute the thermal ensemble average as an analytic function of energy density via kernel Fourier expansion. Implementing this algorithm on a quantum computer has exponential quantum advantage in the cost of time and space, as compared to its classical analogue. In computing finite temperature properties of a quantum system our quantum algorithm has an overall polynomial time complexity, provided that the corresponding Hamiltonian ground state problem belongs to BQP. We demonstrate its efficiency with applications to one and two-dimensional quantum spin models, and a fermionic lattice. By analyzing the realization on digital and analogue quantum devices, we show the quantum algorithm is accessible to near-term quantum technology.

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Algorithms for Gibbs state preparation on noiseless and noisy random quantum circuits

Cited by 6 publications

References 56 publications

Quantum many-body systems in thermal equilibrium

Quantum many-body systems in thermal equilibrium

Quantum algorithms from fluctuation theorems: Thermal-state preparation

Quantum Kernel Function Expansion for Thermal Quantum Ensemble

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