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arXiv:2609.37802

Quantum Complexity of Ancilla-Free Unitary Embeddings for Nonlinear Dynamics via Generalized State-Dependent Double-Bracket Flows

Yuki Ito, Hideaki Hakoshima, Keisuke Fujii

quant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

Simulating nonlinear dynamics with quantum computers has gained increasing attention. In general, such simulations require additional quantum resources because unitary quantum evolution is linear. A fundamental question is how nonlinear dynamics can be embedded into fully coherent, ancilla-free unitary circuits and how the complexity of the dynamics governs the required quantum resources. In this work, we generalize the ancilla-free double-bracket quantum algorithm for imaginary-time evolution by replacing its state-independent Hamiltonian with a state-dependent Hermitian operator. Our framework recursively calls an initial state preparation oracle and its inverse, and prepares the target solution to any prescribed accuracy using a fully coherent, ancilla-free unitary embedding. We relate the query cost to the complexity of the nonlinear dynamics, specifically their sensitivity to initial conditions. We obtain query upper bounds of $\exp(O(T))$, $\exp(O(T^2))$, and $\exp(\exp(O(T)))$ when the distance between solutions contracts at least exponentially (contractive), does not increase (nonexpansive), or grows at most exponentially (expansive), respectively, where $T$ is the target evolution time. For the discrete Gross--Pitaevskii equation, our ancilla-free double-bracket circuit achieves optimal worst-case query complexity $Θ(e^{gT/2})$ over a specified family of single-qubit initial states, where $g>0$ is the nonlinearity strength. These results connect the complexity of nonlinear dynamics to the query cost of coherent quantum simulation and provide a foundation for designing ancilla-free unitary embeddings with optimal query complexity.

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23 pages, 1 figure

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Imaginary-time evolution (ITE) is a widely used technique for preparing the ground state of a target Hamiltonian. Building on the recently proposed framework of Double-Bracket quantum ITE (DB-QITE), we introduce a probabilistic variant (PDBQITE) that achieves an exponential reduction in circuit depth. Our algorithm requires only a single ancilla qubit and a quantum circuit that consists of one controlled real-time evolution and one mid-circuit measurement per iteration. The trade-off is an exponential decay of the total success probability with the number of iterations, for which we derive an analytical lower bound that is independent of system size. Our numerical simulations show that the per-step success probability remains close to unity, making it realistic to execute many iterations, thus providing a substantial improvement over prior constructions. We show that PDBQITE outperforms the near-term probabilistic algorithm PITE for ground-state preparation of molecular Hamiltonians, and that a hybrid QAOA-PDBQITE scheme achieves higher approximation ratios than QAOA at the same circuit depth on random regular graphs for the MaxCut problem.
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This work proposes double-bracket iterations as a framework for obtaining diagonalizing quantum circuits. Their implementation on a quantum computer consists of interlacing evolutions generated by the input Hamiltonian with diagonal evolutions which can be chosen variationally. No qubit overheads or controlled-unitary operations are needed but the method is recursive which makes the circuit depth grow exponentially with the number of recursion steps. To make near-term implementations viable, the proposal includes optimization of diagonal evolution generators and of recursion step durations. Indeed, thanks to this numerical examples show that the expressive power of double-bracket iterations suffices to approximate eigenstates of relevant quantum models with few recursion steps. Compared to brute-force optimization of unstructured circuits double-bracket iterations do not suffer from the same trainability limitations. Moreover, with an implementation cost lower than required for quantum phase estimation they are more suitable for near-term quantum computing experiments. More broadly, this work opens a pathway for constructing purposeful quantum algorithms based on so-called double-bracket flows also for tasks different from diagonalization and thus enlarges the quantum computing toolkit geared towards practical physics problems.
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity O~(νLt/ε)\widetilde{\mathcal{O}}(ν\mathcal{L} t/ε)O(νLt/ε), where the constant ννν depends on the problem parameters, L\mathcal{L}L involves a time integral of upper bounds on the norms of evolution operators, and εεε is the error. In particular, νLν\mathcal{L}νL is linear in ttt for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of Ω(νLt/ε)Ω(ν\mathcal{L} t/ε)Ω(νLt/ε) for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.
Efficiently preparing approximate ground states of large, strongly correlated systems on quantum hardware is challenging, and yet, nature is innately adept at this. This has motivated the study of thermodynamically inspired approaches to ground-state preparation that aim to replicate cooling processes via imaginary-time evolution. However, synthesizing quantum circuits that efficiently implement imaginary-time evolution is itself difficult, with prior proposals generally adopting heuristic variational approaches or using deep block encodings. Here, we use the insight that quantum imaginary-time evolution is a solution of Brockett's double-bracket flow and synthesize circuits that implement double-bracket flows coherently on the quantum computer. We prove that our double-bracket quantum imaginary-time evolution (DB-QITE) algorithm inherits the cooling guarantees of imaginary-time evolution. Concretely, each step is guaranteed to (i) decrease the energy of an initial approximate ground state by an amount proportional to the energy fluctuations of the initial state and (ii) increase the fidelity with the ground state. We provide gate counts for DB-QITE through numerical simulations in qrisp that demonstrate scenarios where DB-QITE outperforms quantum phase estimation. Thus, DB-QITE provides a means to systematically improve the approximation of a ground state using shallow circuits.
For the problem of simulating Lindblad evolution for time ttt to precision εεε, Hamiltonian simulation provides an additive query lower bound, informally, Ω(t+polylog(1/ε))Ω(t + \mathrm{polylog}(1/ε))Ω(t+polylog(1/ε)). However, the best previously known algorithms for general Lindblad simulation achieve a multiplicative upper bound, informally, O(t polylog(1/ε))\mathcal{O}(t\,\mathrm{polylog}(1/ε))O(tpolylog(1/ε)), in query complexity. It has remained open whether this multiplicative dependence is necessary. In this paper, we close the gap in query complexity by giving an algorithm with optimal additive dependence on evolution time and precision in the block-encoding model. Our approach uses the transducer framework to reduce the query cost of composing first-order approximations to the evolution channel, together with linear combinations of reuse circuits of different lengths to suppress catalyst-removal error. We further achieve nearly optimal gate complexity in evolution time and precision through history compression and an efficient implementation of the query-free part of the transducer using operation reordering and linear combinations of unitaries, while preserving the optimal query complexity.