Exponentially accurate open quantum simulation via randomized dissipation with minimal ancilla

TL;DR

The paper develops a quantum algorithm for simulating Lindblad open quantum dynamics with exponentially fast circuit depth for observable estimation, while using only a small, fixed ancilla of size $4+\lceil\log_2 M\rceil$. It achieves this by decomposing the Lindblad propagator $e^{t\mathcal{L}}$ into a linear combination of CPTN superoperators via a transfer-matrix formalism and a Taylor-series-based random compilation that suppresses the norm of the components; a novel exact-dissipation technique recovers the full dynamics with minimal extra qubits. A key contribution is an efficient translation of a linear combination of superoperators into quantum circuits that operate on only $n+1$ qubits, leveraging mid-circuit measurement and a CPTN framework. The algorithm targets estimation of $\mathrm{Tr}[O\rho(t)]$ and requires only $\mathcal{O}(\|O\|^2/\varepsilon^2)$ samples with depth $\mathcal{O}(\log(1/\varepsilon))$, and importantly its gate complexity is independent of the number of jump operators $K$ and the Hamiltonian Pauli-string count $m$. Numerical results on dissipative TFIM and Fermi-Hubbard models demonstrate both accuracy and a favorable resource scaling, highlighting the method’s potential for near-term fault-tolerant quantum devices.

Abstract

Simulating open quantum systems is an essential technique for understanding complex physical phenomena and advancing quantum technologies. Some quantum algorithms for simulating Lindblad dynamics achieve logarithmically short circuit depth in terms of accuracy $\varepsilon$ by coherently encoding all possible jump processes with a large ancilla consumption. Minimizing the space complexity while achieving such a logarithmic depth remains an important challenge. In this work, we present a quantum algorithm for simulating general Lindblad dynamics with multiple jump operators aimed at an observable estimation, that achieves both a logarithmically short circuit depth and a minimum ancilla size. Toward simulating an exponentially accurate Taylor expansion of the Lindblad propagator to ensure the circuit depth of $\mathcal{O}(\log(1/\varepsilon))$, we develop a novel random circuit compilation method that leverages dissipative processes with only a single jump operator; importantly, the proposed method requires the minimal-size, $4 + \lceil \log M \rceil$, ancilla qubits where each single jump operator has at most $M$ Pauli strings. Furthermore, the gate complexity depends on neither the number of terms in Hamiltonian nor the number of jump operators, owing to the random compilation. This work represents a significant step towards making open quantum system simulations more feasible on early fault-tolerant quantum computing devices.

Figures (23)

• Figure 1: Quantum circuit of sampled $\widetilde{\mathcal{W}}_v$ for simulating the decomposition of $e^{t\mathcal{L}}$ in Theorem \ref{['thm: main']}. The colored blocks denote unitary gates, which can be constructed from the input model. The mid-circuit measurement and qubit reset allow us to effectively simulate the CPTN maps $\mathcal{W}_v$ by the quantum circuits. The quantum register $\ket{\bm{0}}$ contains at most $3+\lceil \log_2 M\rceil$ qubits.
• Figure 2: A flowchart for the derivation of Theorem \ref{['thm: main']}. (i) Expansion of the Taylor series of the transfer matrix $e^{tG}$ that can be truncated exponentially accurately. (ii) Decomposition of Eq. \ref{['eq:taylorser']} into type-(A, B) superoperators. For simplicity, the figure shows the flowchart for the dissipation part only. First, we can find the convex combination of minimal dissipative processes $\mathcal{B}_{kl}$ by Eq. \ref{['eq:main_SLL']}. Then, using Lemma \ref{['lemma_main:generalCP']} for the circuit simulation of $\mathcal{B}_{kl}$, we obtain type-(A) superoperators. All other terms including those from Hamiltonian dynamics can be described as type-(B) superoperators. (iii) The desirable linear combination of superoperators $\sum_v c_v S(\mathcal{W}_v)$, obtained by repeating $e^{(t/r)G}$. $\mathcal{W}_v$ is a composite map of type-(A, B) superoperators. (iv) Translation of LCS form $\sum_v c_v S(\mathcal{W}_v)$ into the circuit expression Eq. \ref{['main:dynmapdecomp']}. Lemma \ref{['lem:lcs_simplified']} provides the translation protocol.
• Figure 3: Numerical simulation of the two-level system with the decay. (a) The comparison of the excited state population between the exact solution and the simulation by the proposed algorithm. The exact solution is obtained by QuTiP, and the simulation is obtained with Qiskit density matrix simulation. The error indicates the absolute error between exact and simulation for the single experiment on the right axis. For simplicity, we directly sampled $\mathcal{B}^{\mathrm{(approx)}}$ instead of OAA circuits. (b) The total norm $C$ with respect to $\gamma t$. $C$ is less than $1.5$ at each time $t$ due to the appropriate choice of $r= \max[\lceil 2 \| \mathcal{L} \|_{\mathrm{pauli}}^2 t^2 \rceil ,1 ]$. The upper bound equals to $e$ for $2 \|\mathcal{L}\|_{\mathrm{pauli}}^2 t^2 \ge 1$. (c) Taylor series truncation dependence on the required $\Delta$. This varies in time but is bounded at most 11 for our demonstration. (d) The simulation setup.
• Figure 4: T-gate counts and additional ancilla qubit requirements of our algorithm (this work in red), channel LCU (CLCU in blue) Cleve2016-yj, and first order HS-based (HS in green) Ding2024-SDE for simulating $n$-qubit TFIM and FHM. Since $\|\mathcal{L}\|_{\mathrm{paili}} \propto n$, we employ $\tau/n$ as a horizontal axis instead of $\tau$. We set parameters $(n, \tau/n) = (10^2, 10^2)$ for (a-d), and $(n, \varepsilon) = (10^2, 10^{-4})$ for (e-h).
• Figure 5: The circuit for the generalized Hadamard test. $U_i$ and $U_j$ are the unitary operators randomly sampled.

Theorems & Definitions (34)

• Theorem 1
• Lemma 1: Exact decomposition of general CP maps
• Lemma 2: LCS, simplified
• proof : Sketch of the proof
• Theorem 2
• Remark 1
• Lemma 3
• proof
• Lemma 4: A slightly modified version of Lemma 2 in Ref. Wan2022-tx
• proof : Proof of Theorem \ref{['thm: main']}