Parallel Quantum Algorithm for Hamiltonian Simulation

Abstract

We study how parallelism can speed up quantum simulation. A parallel quantum algorithm is proposed for simulating the dynamics of a large class of Hamiltonians with good sparse structures, called uniform-structured Hamiltonians, including various Hamiltonians of practical interest like local Hamiltonians and Pauli sums. Given the oracle access to the target sparse Hamiltonian, in both query and gate complexity, the running time of our parallel quantum simulation algorithm measured by the quantum circuit depth has a doubly (poly-)logarithmic dependence polylog log(1/ǫ) on the simulation precision ǫ. This presents an exponential improvement over the dependence polylog(1/ǫ) of previous optimal sparse Hamiltonian simulation algorithm without parallelism. To obtain this result, we introduce a novel notion of parallel quantum walk, based on Childs' quantum walk. The target evolution unitary is approximated by a truncated Taylor series, which is obtained by combining these quantum walks in a parallel way. A lower bound Ω(log log(1/ǫ)) is established, showing that the ǫ-dependence of the gate depth achieved in this work cannot be significantly improved.Our algorithm is applied to simulating three physical models: the Heisenberg model, the Sachdev-Ye-Kitaev model and a quantum chemistry model in second quantization. By explicitly calculating the gate complexity for implementing the oracles, we show that on all these models, the total gate depth of our algorithm has a polylog log(1/ǫ) dependence in the parallel setting.

Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications

“…We note that another version of parallel Hamiltonian simulation has been proposed in Ref. [19], achieving doubly logarithmic circuit depth with respect to the precision 𝑂 (log 3 log(1/𝜀)). The algorithm is based on quantum walk and uses a state preparation method with cubic circuit depth.…”

“…In Ref. [19], the authors developed a parallel algorithm to improve the circuit depth to doubly logarithmic dependence on the precision 𝜀. Their protocol is based on the access of two oracles where 𝑖 and 𝑗 are the row and column indexes of Ĥ, 𝑏 is the precision of 𝐻, |𝑧 is a 𝑏-bit basis, ⊕ is the bit-wise XOR, and 𝐿(𝑖, 𝑘) represents the column index of the 𝑘th non-zero element in row 𝑖.…”

“…As an example, we focus on Hamiltonians in the form of Eq. ( 6) in the main text with V𝑙 ( 𝑝) restricted to be Pauli operators (although a more general model has been discussed in [19]). In this case, O 𝑏 H and O L can be realized with circuit depth 𝑂 (𝑛𝑏) and 𝑂 (𝑛) respectively.…”

“…According to Ref. [19], the simulation can be implemented with 𝑂 (𝜏 log(𝛾)) depth of query to O 𝑏 H and O L , 𝑂 (1) depth query to the quantum state preparation unitary for 𝑂 (𝛾) dimensional states, and 𝑂 𝜏 log 2 𝛾 log 2 𝑛 extra depth of single-and two-qubit gates. We have assumed 𝑃 = 𝑂 (poly(𝑛)), and defined 𝜏 ≡ 𝑃𝑡 and 𝛾 ≡ log(𝜏/𝜀).…”

“…Without ancillary qubit, an exponential circuit depth is inevitable to prepare an arbitrary quantum state [6][7][8][9][10][11][12][13][14][15][16] and the optimal result Θ(2 𝑛 /𝑛) is recently obtained by Sun et al [17]. Leveraging ancillary qubits, the circuit depth could be reduced to be sub-exponential scaling [17][18][19][20], yet in the worse case with an exponential number of ancillas. Very recently, the optimal circuit depth Θ(𝑛) was achieved by [17,20] with 𝑂 (2 𝑛 ) [17] and Õ (2 𝑛 ) [20] ancillary qubits.…”