Prepare the thermal Gibbs state of an interacting quantum system on a quantum computer, and use that preparation to evaluate the system's partition function to a target accuracy.
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Prepare the thermal Gibbs state of an interacting quantum system on a quantum computer, and use that preparation to evaluate the system's partition function to a target accuracy. Poulin and Wocjan's algorithm prepares the thermal Gibbs state of an interacting quantum system directly on the quantum register. The paper states that the algorithm sets a universal upper bound D^α on the thermalization time of a quantum system, with D the Hilbert space dimension and the exponent α < 1/2 proportional to the Helmholtz free energy density. A second algorithm derived from the same preparation evaluates the partition function, its running time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared. In the Zoo this sits among methods for approximating ground states, low energy states, and thermal states for some classes of Hamiltonians; the Zoo notes that simulating Hamiltonian time evolution, as well as some problems of preparing ground and thermal states, can all be done as special cases of the quantum singular value transformation.
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Poulin and Wocjan's algorithm prepares the thermal Gibbs state of an interacting quantum system directly on the quantum register. The paper states that the algorithm sets a universal upper bound D^α on the thermalization time of a quantum system, with D the Hilbert space dimension and the exponent α < 1/2 proportional to the Helmholtz free energy density. A second algorithm derived from the same preparation evaluates the partition function, its running time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared. In the Zoo this sits among methods for approximating ground states, low energy states, and thermal states for some classes of Hamiltonians; the Zoo notes that simulating Hamiltonian time evolution, as well as some problems of preparing ground and thermal states, can all be done as special cases of the quantum singular value transformation. This record's speedup class, "Superpolynomial", is a secondary source's classification of the approximation and simulation algorithms it files this under — not a claim its primary paper makes. Not stated by the primary source — abstract of arXiv:0905.2199 was read and makes no such claim. Reported cost: The preparation sets a universal upper bound D^α on the thermalization time, where D is the Hilbert space dimension and α < 1/2 is proportional to the Helmholtz free energy density; the derived partition-function algorithm runs in time proportional to that thermalization time and inversely proportional to the targeted accuracy squared..
ALGORITHM: Thermal Gibbs state sampling
PROBLEM: Prepare the thermal Gibbs state of an interacting quantum system on a quantum computer, and use that preparation to evaluate the system's partition function to a target accuracy.
IDEA: Poulin and Wocjan's algorithm prepares the thermal Gibbs state of an interacting quantum system directly on the quantum register. The paper states that the algorithm sets a universal upper bound D^α on the thermalization time of a quantum system, with D the Hilbert space dimension and the exponent α < 1/2 proportional to the Helmholtz free energy density. A second algorithm derived from the same preparation evaluates the partition function, its running time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared. In the Zoo this sits among methods for approximating ground states, low energy states, and thermal states for some classes of Hamiltonians; the Zoo notes that simulating Hamiltonian time evolution, as well as some problems of preparing ground and thermal states, can all be done as special cases of the quantum singular value transformation.
REPORTED COST: The preparation sets a universal upper bound D^α on the thermalization time, where D is the Hilbert space dimension and α < 1/2 is proportional to the Helmholtz free energy density; the derived partition-function algorithm runs in time proportional to that thermalization time and inversely proportional to the targeted accuracy squared.
BASIS: abstract of arXiv:0905.2199 (plain-text Greek names rendered as Unicode): "This algorithm sets a universal upper bound D^α on the thermalization time of a quantum system, where D is the system's Hilbert space dimension and α < 1/2 is proportional to the Helmholtz free energy density of the system", and "an algorithm to evaluate the partition function of a quantum system in a time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared". The Zoo entry for this section states no bound of its own; its "Superpolynomial" label is the Zoo's classification, not a measured comparison, and is recorded in the speedup field rather than here.
PRIMARY SOURCE: David Poulin, Pawel Wocjan (2009), Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer — https://arxiv.org/abs/0905.2199
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Primary source: it states the Gibbs-state preparation for interacting quantum systems, the universal D^α thermalization-time bound with α < 1/2 tied to the Helmholtz free energy density, and the partition-function algorithm's dependence on thermalization time and on the squared target accuracy.
arxiv.org/abs/0905.2199 ↗