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Complexity of Implementing Trotter Steps
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DOI：10.1103/PRXQuantum.4.020323.png)
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Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner. However, such an implementation of Trotter steps has gate com-plexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques. We develop methods to perform faster Trotter steps with complexity sub-linear in the number of terms. We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law. Our methods include one based on a recursive block encod-ing and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques. We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank. Combining with a tighter error analysis, we show that it suf-fices to use (771/3n1/3 + n2/3/772/3) n1+o(1) gates to simulate uniform electron gas with n spin orbitals and 77 electrons in second quantization in real space, asymptotically improving over the best previous work. We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation. We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic n-qubit two-local Hamiltonians with commuting terms require at least S2 (n2) gates to evolve with accuracy e = S2 (1/poly(n)) for time t = S2 (e). Our proof is based on a gate-efficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-2 subspace, which may be of independent interest. Our result thus suggests the use of Hamiltonian struc-tural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.
Keyword:
QUANTUM
SIMULATION
ALGORITHMS
MATRICES
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IF:
11
论文数:
919
被引数:
9.0K
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