Quantum Hamiltonian complexity in thermal equilibrium

The physical properties of a quantum many-body system in thermal equilibrium are determined by its partition function and free energy. Here we study the computational complexity of approximating these quantities for n -qubit local Hamiltonians. First, we report a classical algorithm with poly( n ) r...

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Published in:	Nature physics Vol. 18; no. 11; pp. 1367 - 1370
Main Authors:	Bravyi, Sergey, Chowdhury, Anirban, Gosset, David, Wocjan, Pawel
Format:	Journal Article
Language:	English
Published:	London Nature Publishing Group UK 01.11.2022 Nature Publishing Group
Subjects:	639/766/483/481 639/766/530/2804 639/766/530/951 Algorithms Approximation Atomic Circuits Classical and Continuum Physics Complex Systems Complexity Condensed Matter Physics Energy Equilibrium Free energy Hamiltonian functions Ising model Mathematical analysis Mathematical and Computational Physics Molecular Optical and Plasma Physics Optimization Partitions (mathematics) Physical properties Physics Physics and Astronomy Polynomials Quantum computing Qubits (quantum computing) Run time (computers) Theoretical
ISSN:	1745-2473, 1745-2481
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Abstract	The physical properties of a quantum many-body system in thermal equilibrium are determined by its partition function and free energy. Here we study the computational complexity of approximating these quantities for n -qubit local Hamiltonians. First, we report a classical algorithm with poly( n ) runtime, which approximates the free energy of a given 2-local Hamiltonian provided that it satisfies a certain denseness condition. Our algorithm contributes to a body of work investigating the hardness of approximation for difficult optimization problems. Specifically, this extends existing efficient approximation algorithms for dense instances of the ground energy of 2-local quantum Hamiltonians and the free energy of classical Ising models. Second, we establish polynomial-time equivalence between the problem of approximating the free energy of local Hamiltonians and several other natural tasks ubiquitous in condensed-matter physics and quantum computing, such as the problem of approximating the number of input states accepted by a polynomial-size quantum circuit. These results suggest that the simulation of quantum many-body systems in thermal equilibrium may precisely capture the complexity of a broad family of computational problems that have yet to be defined or characterized in terms of known complexity classes. Finally, we summarize state-of-the-art classical and quantum algorithms for approximating the free energy and show how to improve their runtime and memory footprint. A quantum many-body system’s equilibrium behaviour is described by its partition function, which is hard to compute. Now it has been shown that the easier task of finding an approximation could define a distinct class of computational problems.
AbstractList	The physical properties of a quantum many-body system in thermal equilibrium are determined by its partition function and free energy. Here we study the computational complexity of approximating these quantities for n-qubit local Hamiltonians. First, we report a classical algorithm with poly(n) runtime, which approximates the free energy of a given 2-local Hamiltonian provided that it satisfies a certain denseness condition. Our algorithm contributes to a body of work investigating the hardness of approximation for difficult optimization problems. Specifically, this extends existing efficient approximation algorithms for dense instances of the ground energy of 2-local quantum Hamiltonians and the free energy of classical Ising models. Second, we establish polynomial-time equivalence between the problem of approximating the free energy of local Hamiltonians and several other natural tasks ubiquitous in condensed-matter physics and quantum computing, such as the problem of approximating the number of input states accepted by a polynomial-size quantum circuit. These results suggest that the simulation of quantum many-body systems in thermal equilibrium may precisely capture the complexity of a broad family of computational problems that have yet to be defined or characterized in terms of known complexity classes. Finally, we summarize state-of-the-art classical and quantum algorithms for approximating the free energy and show how to improve their runtime and memory footprint.A quantum many-body system’s equilibrium behaviour is described by its partition function, which is hard to compute. Now it has been shown that the easier task of finding an approximation could define a distinct class of computational problems. The physical properties of a quantum many-body system in thermal equilibrium are determined by its partition function and free energy. Here we study the computational complexity of approximating these quantities for n -qubit local Hamiltonians. First, we report a classical algorithm with poly( n ) runtime, which approximates the free energy of a given 2-local Hamiltonian provided that it satisfies a certain denseness condition. Our algorithm contributes to a body of work investigating the hardness of approximation for difficult optimization problems. Specifically, this extends existing efficient approximation algorithms for dense instances of the ground energy of 2-local quantum Hamiltonians and the free energy of classical Ising models. Second, we establish polynomial-time equivalence between the problem of approximating the free energy of local Hamiltonians and several other natural tasks ubiquitous in condensed-matter physics and quantum computing, such as the problem of approximating the number of input states accepted by a polynomial-size quantum circuit. These results suggest that the simulation of quantum many-body systems in thermal equilibrium may precisely capture the complexity of a broad family of computational problems that have yet to be defined or characterized in terms of known complexity classes. Finally, we summarize state-of-the-art classical and quantum algorithms for approximating the free energy and show how to improve their runtime and memory footprint. A quantum many-body system’s equilibrium behaviour is described by its partition function, which is hard to compute. Now it has been shown that the easier task of finding an approximation could define a distinct class of computational problems.
Author	Gosset, David Chowdhury, Anirban Wocjan, Pawel Bravyi, Sergey
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Snippet	The physical properties of a quantum many-body system in thermal equilibrium are determined by its partition function and free energy. Here we study the...
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SubjectTerms	639/766/483/481 639/766/530/2804 639/766/530/951 Algorithms Approximation Atomic Circuits Classical and Continuum Physics Complex Systems Complexity Condensed Matter Physics Energy Equilibrium Free energy Hamiltonian functions Ising model Mathematical analysis Mathematical and Computational Physics Molecular Optical and Plasma Physics Optimization Partitions (mathematics) Physical properties Physics Physics and Astronomy Polynomials Quantum computing Qubits (quantum computing) Run time (computers) Theoretical
Title	Quantum Hamiltonian complexity in thermal equilibrium
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