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Bulletin of the American Physical Society

Quantum Integrability in Generic Time Dependent Hamiltonians

Nikolai A. Sinitsyn, Emil A. Yuzbashyan, Vladimir Chernyak, Aniket Patra, Chen Sun

Published Mar 8, 2018• 0 citations
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Product formula (PF), which approximates the time evolution under a many-body Hamiltonian by the product of local time-evolution operators, is one of the central approaches for simulating quantum dynamics by quantum computers. It has been of great interest whether the errors of PFs can be expressed by commutators among local terms, since it brings the substantial suppression of the computational cost in the system size reflecting the locality of Hamiltonians. Although recent studies have revealed such error bounds for generic time-independent systems, it remains to be a difficult problem for time-dependent Hamiltonians except for low-order PFs. In this paper, we derive an error bound of generic PFs for smooth time-dependent Hamiltonians, which is expressed explicitly by nested commutators and their time derivatives. It indicates the substantial error suppression in the system size for generic local Hamiltonians, thereby giving a much better estimate of gate counts. Our derivation employs Floquet theory originally developed for time-periodic systems, with which we analyze the time-dependent PF errors by mapping them to those for time-independent PFs defined on an infinite-dimensional space. This approach is also applicable to a family of time-dependent PFs, such as the one for non-unitary dynamics. In particular, we clarify the error bound of a time-dependent multi-product formula, with which we can achieve exponentially better scaling with respect to the target accuracy than the PF. Our results will shed light on applications of near-term and future quantum computers, ranging from quantum simulation of nonequilibrium materials to faster algorithms exploiting time-dependent Hamiltonians such as adiabatic state preparation.
Simulating Hamiltonian dynamics is one of the most fundamental and significant tasks for characterising quantum materials. Recently, a series of quantum algorithms employing block-encoding of Hamiltonians have succeeded in providing efficient simulation of time-evolution operators on quantum computers. While time-independent Hamiltonians can be simulated by the quantum eigenvalue transformation (QET) or quantum singularvalue transformation with the optimal query complexity in time ttt and desirable accuracy ε\varepsilonε, generic time-dependent Hamiltonians face at larger query complexity and more complicated oracles due to the difficulty of handling time-dependency. In this paper, we establish a QET-based approach for simulating time-dependent Hamiltonians with multiple time-periodicity. Such time-dependent Hamiltonians involve a variety of nonequilibrium systems such as time-periodic systems (Floquet systems) and time-quasiperiodic systems. Overcoming the difficulty of time-dependency, our protocol can simulate the dynamics under multi-periodic time-dependent Hamiltonians with optimal/nearly-optimal query complexity both in time ttt and desirable accuracy ε\varepsilonε, and simple oracles as well as the optimal algorithm for time-independent cases.
Simulating Hamiltonian dynamics is one of the most fundamental and significant tasks for characterizing quantum materials. Recently, a series of quantum algorithms employing block encoding of Hamiltonians have succeeded in providing efficient simulation of time-evolution operators on quantum computers. While time-independent Hamiltonians can be simulated by the quantum eigenvalue transformation (QET) or quantum singular value transformation with the optimal query complexity in time ttt and desirable accuracy $\ensuremath{\varepsilon}$, generic time-dependent Hamiltonians face at larger query complexity and more complicated oracles due to the difficulty of handling time-dependency. In this paper, we establish a QET-based approach for simulating time-dependent Hamiltonians with multiple time periodicity. Such time-dependent Hamiltonians involve a variety of nonequilibrium systems such as time-periodic systems (Floquet systems) and time-quasiperiodic systems. Overcoming the difficulty of time dependency, our protocol can simulate the dynamics under multiperiodic time-dependent Hamiltonians with optimal and nearly optimal query complexity both in time ttt and desirable accuracy $\ensuremath{\varepsilon}$, and simple oracles as well as the optimal algorithm for time-independent cases.