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Lower bound for simulation cost of open quantum systems: Lipschitz continuity approach

Zhiyan Ding, Marius Junge, Philipp Schleich, Peixue Wu

TL;DR

The paper addresses the problem of lower-bounding the quantum circuit depth needed to simulate open quantum systems governed by Lindblad dynamics under a fixed set of accessible unitaries. It introduces a convexified, randomized-scheme cost (convexified circuit depth) and a Lipschitz-complexity framework C_S^{cb} to translate channel-approximation errors into mandatory simulation depth, yielding a universal lower bound in terms of the ergodic average E_{\infty} and the mixing time. For unital dynamics, the authors establish lower bounds of the form M_{\alpha,\beta} \ge C_{\alpha,\beta} \kappa(S)/t_{mix}, and show tightness in concrete Pauli-noise models, with complementary upper bounds from Trotter and Poisson schemes. For non-unital dynamics, they demonstrate that environment-assisted simulation is essential, develop a compatible Lipschitz-complexity theory on the enlarged system, and provide both general lower bounds and concrete bounds for amplitude-damping and related noise models. The results highlight a dimension-dependent factor and show that, in important cases, upper and lower bounds match up to constants, guiding the design and assessment of open-system quantum simulations with constrained hardware. The work has implications for evaluating quantum-simulation efficiency, especially when environment effects cannot be neglected, and offers a principled approach to quantify inherent limits of simulation schemes.

Abstract

Simulating quantum dynamics is one of the most promising applications of quantum computers. While the upper bound of the simulation cost has been extensively studied through various quantum algorithms, much less work has focused on establishing the lower bound, particularly for the simulation of open quantum system dynamics. In this work, we present a general framework to calculate the lower bound for simulating a broad class of quantum Markov semigroups. Given a fixed accessible unitary set, we introduce the concept of convexified circuit depth to quantify the quantum simulation cost and analyze the necessary circuit depth to construct a quantum simulation scheme that achieves a specific order. Our framework can be applied to both unital and non-unital quantum dynamics, and the tightness of our lower bound technique is illustrated by showing that the upper and lower bounds coincide in several examples.

Lower bound for simulation cost of open quantum systems: Lipschitz continuity approach

TL;DR

The paper addresses the problem of lower-bounding the quantum circuit depth needed to simulate open quantum systems governed by Lindblad dynamics under a fixed set of accessible unitaries. It introduces a convexified, randomized-scheme cost (convexified circuit depth) and a Lipschitz-complexity framework C_S^{cb} to translate channel-approximation errors into mandatory simulation depth, yielding a universal lower bound in terms of the ergodic average E_{\infty} and the mixing time. For unital dynamics, the authors establish lower bounds of the form M_{\alpha,\beta} \ge C_{\alpha,\beta} \kappa(S)/t_{mix}, and show tightness in concrete Pauli-noise models, with complementary upper bounds from Trotter and Poisson schemes. For non-unital dynamics, they demonstrate that environment-assisted simulation is essential, develop a compatible Lipschitz-complexity theory on the enlarged system, and provide both general lower bounds and concrete bounds for amplitude-damping and related noise models. The results highlight a dimension-dependent factor and show that, in important cases, upper and lower bounds match up to constants, guiding the design and assessment of open-system quantum simulations with constrained hardware. The work has implications for evaluating quantum-simulation efficiency, especially when environment effects cannot be neglected, and offers a principled approach to quantify inherent limits of simulation schemes.

Abstract

Simulating quantum dynamics is one of the most promising applications of quantum computers. While the upper bound of the simulation cost has been extensively studied through various quantum algorithms, much less work has focused on establishing the lower bound, particularly for the simulation of open quantum system dynamics. In this work, we present a general framework to calculate the lower bound for simulating a broad class of quantum Markov semigroups. Given a fixed accessible unitary set, we introduce the concept of convexified circuit depth to quantify the quantum simulation cost and analyze the necessary circuit depth to construct a quantum simulation scheme that achieves a specific order. Our framework can be applied to both unital and non-unital quantum dynamics, and the tightness of our lower bound technique is illustrated by showing that the upper and lower bounds coincide in several examples.
Paper Structure (26 sections, 29 theorems, 248 equations, 6 figures)

This paper contains 26 sections, 29 theorems, 248 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related works and open questions
  4. Preliminaries on simulation cost of quantum channels
  5. Upper estimates for uniform maximal simulation cost of unital quantum dynamics.
  6. Symmetric noise generators: Trotter approximation
  7. Unitary noise generators: Poisson model
  8. Lower estimates for uniform maximal simulation cost of unital quantum dynamics
  9. Generic lower bound estimates via Lipschitz complexity
  10. Preliminary on Lipschitz complexity
  11. Generic lower bound approach: Lipschitz continuity.
  12. Lower bound with fixed target time and precision
  13. An illustrating example: $n$-qubit Pauli noise
  14. Trotter approach
  15. Poisson approach
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose the accessible unitary gate set $\mathcal{U}$ is given. Under the ergodic assumption and that we can choose a compatible resource set $S$ (see Definition compatible: U and S and the discussion after that), the lower bound of $\mathrm{M}_{\alpha,\beta}$ is given as where $C_S^{cb}(E_{\infty})$ is a convex cost function of the quantum channel $E_{\infty}$, defined in Definition eqn:C_cb_ph

Figures (6)

  • Figure 1: Unital noise models can be treated without allocating an explicit environment as quantum states, which is described in Sections \ref{['sec:upper']} and \ref{['sec:lower_bound']}. The case of non-unital noise with environment and correspondingly adjusted Lipschitz complexity is discussed in Section \ref{['sec:non-unital']}.
  • Figure 2: One iteration of simulating non-unital quantum dynamics with support of an environment $E$. Around the actual simulation, there is an encoding and a decoding map. This is discussed in more detail in Section \ref{['sec:non-unital']}.
  • Figure 3: Overview of determining lower bounds of noise models using Lipschitz complexity. We are given simulation parameters $\alpha,\beta>0$, a resource set of bounded operators and a set of admissible unitary gates $\mathcal{U}$ so that every gate in the set has at most complexity $D$ under the Lipschitz complexity induced by the resource set ($D$-compatible). Then, the lower bound can be determined as a function of the resource-set induced mixing time and the expected length $\kappa(S)$, which both rely on knowledge about the conditional expectation on fixed-point densities $E_\infty$ of the open system channel.
  • Figure 4: Quantum circuit for non-unital quantum dynamics simulation (a single step).
  • Figure 5: Using a natural amplified accessible gate set $\mathcal{U}_{A^{\otimes n} \otimes E} = \{\pi_j(u_{AE}): u_{AE} \in \mathcal{U}_{A\otimes E}, 1\le j \le n\}$ to approximate $\mathcal{T}_t^{\otimes n}$.
  • ...and 1 more figures

Theorems & Definitions (70)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 60 more