Continuous-time evolution via probabilistic angle interpolation and its applications

The gist: This paper explores a stochastic time-evolution algorithm based on probabilistic angle interpolation to simulate continuous-time evolution, simplifying pre-processing by taking the continuous-time limit and introducing noise mitigation tailored to it.

Stochastic Time Evolution Algorithm (TE-PAI)

The core of the method is the stochastic time-evolution algorithm based on probabilistic angle interpolation (TE-PAI) [9, 11]. This algorithm realizes channels representing the Trotterized time evolution without time-discretization error after averaging over stochastically sampled unitary circuits [9, 11]. The key ingredient involves the identity for the unitary channel Rk(θk)[ρ] which is given by Rk(θk) = γ1(θk)I + γ2(θk)Rk(sgn(θk)∆) + γ3(θk)Rk(π) [11]. The protocol constructs an estimator of Uτ(t) as Uˆτ(t) = Y R r=1 Y K k=1 Rˆ r k (3).

Continuous-Time Limit and Simplification

To simplify the pre-processing, the authors explicitly take the continuous-time limit τ → 0, which eliminates Trotter error and simulates Hamiltonian evolution without bias [2]. This leads to the sampling rates r(k)2 = lim τ→0 p(k)2τ = 2cksin ∆ and r(k)3 = lim τ→0 p(k)3τ = ck tan ∆ squared for each Hamiltonian term Pk [6]. The total normalization factor, i.e., sampling overhead, is given by Nsample = lim τ→0 Y k 1 + 2τck tan ∆ squared + O(τ 2)!t/τ = exp 2t∥c∥1 tan ∆ squared , with∥c∥1:= P k ck [7]. The average number of gates for each circuit is Ngates:= t∥c∥1 2 sin ∆ + tan ∆ 2! = t∥c∥1 3 − cos ∆ sin ∆ [8].

Error Mitigation: Zero-Noise Extrapolation (ZNE)

The authors tailor the zero-noise extrapolation (ZNE) to the stochastic algorithm by selecting two angle values, ∆1 and ∆2, and running the protocol with both angles to obtain expectation values y1 and y2 [9]. Assuming Ngates ≈ 2t∥c∥1/∆ for simplicity, a linear extrapolation is performed assuming that the effect of noise on the expectation values is linear in the number of gates, yi = a + bNgates(∆i), with fitting parameters a, b [14]. The mitigated expectation value is obtained by extrapolating y to the ∆ → ∞ limit as yZNE = ∆1y1 − ∆2y2 / ∆1 − ∆2 [9].

Applications

The protocol was demonstrated on two representative problems:

1. Estimating the ground-state energy of the H+3 molecular Hamiltonian using adiabatic ground-state preparation and the Hadamard test [9, 15]. This involved preparing an approximate ground state using TE-PAI and then applying TETRIS to stochastically implement the Hadamard test [9, 7]. The resulting estimate is given by Eest = EHF + 1/s arctan tan(sϵ) η+ + η− η+ − η-! [16, 14].

2. Computing out-of-time-ordered correlators (OTOCs) in the sparse Sachdev–Ye–Kitaev (SYK) model by classically simulating noisy quantum circuits [9]. The OTOC is computed by taking the weighted average of Uˆ1 and Uˆ2 according to Eq. (4), which recovers Re[OTOCi, j] [17].

Hardware Demonstration and Discussion

The protocol was evaluated on a trapped-ion quantum computer, Quantinuum Reimei [9]. For the H+3 simulation, the stochastic algorithm estimates the energy without bias, albeit with sampling error, reaching accurate values with moderate two-qubit gate counts [14]. For sparse SYK OTOCs on Reimei, noisy simulation shows that extrapolation across ∆ values can partially recover the noiseless trend [9]. The experimental demonstration on Reimei exhibits a similar trend but the vanishing signal due to hardware noise makes it challenging to suppress the statistical error [9]. The main practical challenge is improving the balance between sampling cost and hardware noise by adapting scheduling of ∆ over evolution time [9, 32–35].

Algorithm Details

The adiabatic evolution circuit is constructed by drawing times and angles according to Poisson processes of rates r(k)2 and r(k)3 [8]. The TETRIS algorithm constructs an operator Vˆ H that approximates VH on average without bias, NTETE[Vˆ H] = VH, at the cost of the sampling overhead NTET = e s∥a∥1 tan ∆ squared with a free parameter ∆ > 0 [9]. The final energy estimate is read off via Im e isEHF ⟨HF U† AdVHUAd HF⟩ ≈ sin[s(Eexact − E± HF)] [24]. The algorithm's tunability, where the angle parameter ∆ controls circuit depth through Eq. (8) and sampling overhead through Eq. (7), enables a stochastic-ZNE strategy by comparing runs at different ∆ values [9]. The paper concludes that the continuous-time TE-PAI directly eliminates Trotter bias at the channel level while simplifying classical pre-processing [9].

--- Page 1 ---

RIKEN-iTHEMS-Report-26 Continuous-time evolution via probabilistic angle interpolation and its applications Tomoya Hayata1, 2, 3, ∗ and Yuta Kikuchi4, 2, † 1Departments of Physics, Keio University School of Medicine, 4-1-1 Hiyoshi, Kanagawa 223-8521 Japan

RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN, Wako 351-0198 Japan

International Center for Elementary Particle Physics and The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 Japan

(Dated: April 6, 2026) We explore the applicability of a stochastic time-evolution algorithm based on probabilistic angle interpolation. To simplify the pre-processing of the algorithm, we take the continuous-time limit, thereby explicitly eliminating Trotter errors and streamlining the resource analysis. We also introduce a noise-mitigation method tailored to it. As demonstrations, we apply the algorithm to two representative problems: estimating the ground-state energy of the H+3 molecular Hamiltonian and computing out-of-time-ordered correlators in the sparse Sachdev–Ye–Kitaev model. We evaluate the protocol’s performance through numerical simulations and experiments on a trapped-ion quantum computer, Quantinuum Reimei.

--- Page 2 ---

2 As discussed in [9], the step size τ can be made small to achieve an arbitrarily small time-discretization error without increasing the circuit size, which in turn introduces R = t/τ classical overhead to construct each stochastically sampled circuit. Inspired by [7], we simplify the pre-processing by explicitly taking the continuous-time limit τ → 0, which also allows us to eliminate the Trotter error, and simulate the Hamiltonian evolution without bias. Introducing the rates of sampling (γ2,Rk(sgn(θk)∆)) and (γ3,Rk(π)), respectively, r(k)2= lim τ→0 p(k)2τ = 2cksin ∆, r(k)3= lim τ→0 p(k)3τ = ck tan ∆ squared, for each Hamiltonian term Pk, we propose the following protocol to approximate the time evolution for the time t: 1) For each k ∈ [1,..., K], draw t(k)1,..., t(k)m and s(k)1,..., s(k)m according to Poisson processes of rates r(k)2 and r(k)3, respectively. 2) Assign the time t (k)m to Rk(sgn(θk∆) and s (k)m to Rk(π), and apply the channels to an input state in the increasing order of the associated times. Below, we denote the constructed estimator and the corresponding unitary operator by Uˆ (t):= limτ→0 Uˆτ(t) and Uˆ (t), respectively.

--- Page 3 ---

3 Since the renormalization factor for each step is given by (5) before taking the continuous-time limit, i.e., sampling overhead, is Nsample = lim τ→0 Y k 1 + 2τck tan ∆ squared + O(τ 2)!t/τ = exp 2t∥c∥1 tan ∆ squared , with∥c∥1:= P k ck. Using the gate sampling rates (6), one finds the average number of gates for each circuit Ngates:= t∥c∥1 2 sin ∆ + tan ∆ 2! = t∥c∥1 3 − cos ∆ sin ∆.

1. The system can estimate ground-state energy of molecular Hamiltonians (like H+3) using a stochastic time-evolution algorithm that achieves unbiased energy estimate by combining TE-PAI with TETRIS, which is NTETE[Vˆ H] = VH with a known normalization factor NTET.

2. The system can compute out-of-time-ordered correlators (OTOCs) in sparse Sachdev–Ye–Kitaev (SYK) models by employing the interferometric circuit and ZNE, which recovers Re[OTOCi, j] from sampled TE-PAI circuits.

3. The system can perform noisy quantum simulation of complex Hamiltonians and OTOCs with noise mitigation via zero-noise extrapolation (ZNE), allowing for the estimation of physical quantities even when hardware noise is present, as shown by the results in Figure 4 and 6.

We explore the applicability of a stochastic time-evolution algorithm based on probabilistic angle interpolation. To simplify the pre-processing of the algorithm, we take the continuous-time limit, thereby explicitly eliminating Trotter errors and streamlining the resource analysis. We also introduce a noise-mitigation method tailored to it. As demonstrations, we apply the algorithm to two representative problems: estimating the ground-state energy of the H 3+ molecular Hamiltonian and computing out-of-time-ordered correlators in the sparse Sachdev--Ye--Kitaev model. We evaluate the protocol's performance through numerical simulations and experiments on a trapped-ion quantum computer, Quantinuum Reimei.