Pith. sign in

REVIEW 3 major objections 4 minor 85 references

Simulating quantum collision models with Hamiltonian simulations using early fault-tolerant quantum computers

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Open-system dynamics can be simulated with collision models, a single ancilla, and no block encodings.

desk verdict Solid algorithmic framework with a genuinely useful LCU-composition trick, but the central depth formula is inverted in the theorem and tables—fixable but currently wrong. read the letter →

arxiv 2504.21564 v2 pith:WA2BNMG2 submitted 2025-04-30 quant-ph cs.DS

Kushagra Garg , Zeeshan Ahmed , Subhadip Mitra , Shantanav Chakraborty This is my paper · ORCID
classification quant-phcs.DS MSC 81P6881S22 PACS 03.67.Ac03.65.Yz
keywords quantumcollisionmodelsopensystemsLindbladmasterequationHamiltoniansimulationsingle-ancillaLCUqDRIFTnon-Markoviandynamicsearlyfault-tolerantcomputing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper attempts to establish that quantum collision models—repeated short interactions between a system and a stream of environment qubits—can simulate open-system dynamics on early fault-tolerant quantum computers using only near-term Hamiltonian simulation routines, without block-encoded oracles and with at most $n+2$ qubits. The authors prove that composing $K$ approximate Hamiltonian evolutions and tracing out the environment after each step yields an $\varepsilon$-accurate estimate of the expectation value of an observable, provided each single-evolution simulation is accurate to $\varepsilon/(3K\|O\|)$. They then use the known correspondence between memoryless collisions and Lindblad dynamics to give an end-to-end algorithm for simulating $e^{\mathcal{L}t}$, with classical repetitions scaling as $O(\|O\|^2 \log(1/\delta)/\varepsilon^2)$ and per-run circuit depth depending on the chosen Hamiltonian simulation method. They compare first- and higher-order Trotterization, qDRIFT, and a single-ancilla linear combination of unitaries (SA-LCU), finding SA-LCU's depth scales as $\tilde{O}(m^3 t^3 \|O\|/(\varepsilon\, \Gamma \beta_{\max}^2))$ and numerically that it dominates the other near-term methods at high precision on a 10-qubit spin chain. The same construction extends to memory-retaining collisions with partial swaps between environment qubits, giving non-Markovian simulations with similar costs.

What carries the argument

The load-bearing object is the Markovian $K$-collision map $\Phi_j[\cdot]=\mathrm{Tr}_{E_j}[U_j(\cdot\otimes \rho_{E_j})U_j^\dagger]$ with $U_j=e^{-i\beta_j H_j \Delta t}$, composed $K$ times, and its implementation by a single-ancilla LCU called SA-LCU. Each $U_j$ is expanded as an approximate LCU $\sum_k \alpha_{jk} W_{jk}$; at every collision two unitaries $X_j$ and $Y_j$ are independently sampled from the distribution $(W_{jk}, \alpha_{jk}/\alpha^{(j)})$, applied as controlled and anti-controlled gates on a single ancilla, and the environment register is traced out. The product $\zeta=\prod_j \alpha^{(j)}$ of LCU weights is kept $O(1)$ by choosing the LCU parameter $r=O(\beta^2 \Delta t^2 K)$, which prevents the naive exponential growth $\alpha^K$, and the final measurement of $\sigma_x\otimes O$ has expectation $\mathrm{Tr}[O M_K[\rho_S]]/\zeta^2$. This sampling-and-trace machinery is what turns any near-term Hamiltonian simulation subroutine into a collision-model simulator with only one ancilla qubit.

What would settle it

Take the simplest nontrivial case—a single qubit with amplitude damping, $\mathcal{L}[\rho]=\gamma(\sigma_-\rho\sigma_+ - \frac{1}{2}\{\sigma_+\sigma_-,\rho\})$—and numerically compute the induced 1-norm distance $\|e^{\mathcal{L}t} - M_{m,\nu}\|_{1\to 1}$ for a range of $t$, $m$, and $\varepsilon$, finding the smallest $\nu$ needed for $\varepsilon$-accuracy. If the minimal $\nu$ does not grow as $t^2 m \Gamma/\varepsilon$ (or the constant is not benign), the claimed $\tilde{O}(m^3 t^3/\varepsilon)$ circuit depth for SA-LCU must be revised. The same numerical experiment can be run with the paper's full circuits on a classical simulator for $n=2$ or $3$ qubits to verify the $\varepsilon$-additive output of Algorithm 2.

Watch

Extended reading notes

Core claim

The central claim is that simulating Lindbladian dynamics does not require specialized block-encoding oracles or many ancillas: a Lindblad generator $e^{\mathcal{L}t}$ can be approximated by an $(m,\nu)$-collision map, and that map can be implemented by recycling a single environment register plus one ancilla through $K=m\nu$ controlled Hamiltonian evolutions, drawing each controlled unitary at random from an LCU decomposition of the collision time-evolution. Provided each collision unitary is simulated to precision $\varepsilon/(3K\|O\|)$, the algorithm outputs an $\varepsilon$-additive estimate of $\mathrm{Tr}[O e^{\mathcal{L}t}[\rho_S]]$ with probability $1-\delta$ in $T=O(\|O\|^2 \log(1/\delta)/\varepsilon^2)$ coherent runs. For the SA-LCU subroutine the per-run circuit depth is $\tilde{O}(m^3 t^3 \|O\|/(\varepsilon\,\Gamma\beta_{\max}^2))$, which the paper shows is better in precision than first-order Trotter and qDRIFT and competitive with second-order Trotter depending on $t$ and $\varepsilon$; the numerical benchmark on a ten-qubit transverse-field Ising chain under amplitude damping shows SA-LCU needing about 200 times fewer CNOT gates than second-order Trotter and 2000 times fewer than qDRIFT at $\varepsilon=10^{-4}$. A parallel result for non-Markovian collisions shows that inserting partial-swap channels between consecutive environment qubits preserves the same per-collision error budget and yields a CPTP non-Markovian $K$-collision map with the same asymptotic circuit depth.

Load-bearing premise

The load-bearing premise is a bound, taken from another paper, that a collision model with $\nu=O(t^2 m \Gamma/\varepsilon)$ repetitions is within $\varepsilon$ of the true Lindblad evolution in induced 1-norm; the authors rely on that lemma without proof, and if its true parameter dependence differs, every resource count in the paper changes.

Editorial extensions

If this is right

  • Until error-corrected machines with large ancilla budgets arrive, Lindblad dynamics for $n$-qubit systems can be simulated on early fault-tolerant devices with at most $n+2$ qubits, no block encodings, and no multi-controlled oracle logic.
  • The same framework can simulate non-Markovian dynamics by adding partial-swap channels between environment qubits, so memory effects are accessible at circuit depths comparable to the Markovian case.
  • At high precision the single-ancilla LCU method gives shorter circuits than first-order Trotter and qDRIFT; at long evolution times second-order Trotter becomes competitive, so method choice should depend on the target $\varepsilon$ and $t$.
  • The $T=O(\|O\|^2 \log(1/\delta)/\varepsilon^2)$ sample count is independent of the Hamiltonian simulation subroutine, so the usual Monte Carlo repetition overhead for expectation estimation is not worsened by the collision framework.
  • Because the per-collision simulation error budget is $\varepsilon/(3K\|O\|)$, any future Hamiltonian simulation algorithm with better depth scaling can be dropped into the framework and immediately improves the end-to-end collision simulation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct numerical check of the cited $\nu=O(t^2 m \Gamma/\varepsilon)$ bound on small qubit systems would determine whether the collision-model approach is practical at the quoted depths; this is not done in the paper.
  • The non-Markovian section proves correctness for finite $K$ but does not give the finite-$K$ distance to a continuous non-Markovian master equation; deriving such a bound would turn the framework into an end-to-end non-Markovian simulator in the same sense as the Lindblad case.
  • Because the per-collision precision requirement is linear in $K$, extrapolation or error-mitigation techniques could be layered on top of Trotter or qDRIFT to reduce the required $K$, potentially making those subroutines competitive with SA-LCU at high precision.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript develops randomized quantum algorithms for simulating quantum collision models on early fault-tolerant quantum computers. It presents a general framework in which a K-collision Markovian map is implemented by composing Hamiltonian simulations, and then specializes to Lindbladian dynamics via an (m,ν)-collision map, using a bound from Pocrnic et al. to set ν. The authors compare circuit depths for first-, second-, and higher-order Trotterization, qDRIFT, and single-ancilla LCU (SA-LCU), claiming that SA-LCU achieves circuit depth Õ(m³t³‖O‖/(ε Γ βmax²)) while using at most n+2 qubits and no block encodings. They also benchmark CNOT counts on a 10-qubit transverse-field Ising model under amplitude damping, and extend the framework to non-Markovian collisions using partial swaps between environment registers.

Significance. If the resource formulas were correct, the paper would provide a useful qubit-efficient, block-encoding-free route to simulating open-system dynamics via collision models in the early-fault-tolerant regime, and its comparison across Hamiltonian simulation subroutines would be a valuable reference. Strengths include a modular proof structure (Theorem 1, Theorem A3), an explicit re-derivation of the SA-LCU decomposition in Appendix B, and an end-to-end construction that avoids amplitude estimation and block encodings. The reliance on Lemma 3 from Ref. [55] is explicit and not circular. However, the central quantitative claim contains an internal inversion inconsistency, so the asymptotic and numerical comparisons are not currently supported. The numerical benchmark is also not reproducible from the text alone.

major comments (3)
  1. [Theorem 2, Eq. (57) and Eq. (64)] The circuit depth stated in Theorem 2 does not follow from the proof. Starting from Eq. (43) with K=mν, Δt=t/ν, β≤O(sqrt(ν/t) βmax), and ν=O(t²m‖O‖Γ/ε), the proof obtains τd = O(m³t³βmax²‖O‖Γ/ε · polylog) in Eq. (63). The theorem then states τd = Õ(m³t³‖O‖/(ε Γ βmax²)), which inverts both Γ and βmax. Since ν is proportional to Γ, a larger Γ means more collision blocks and should increase the circuit depth, not decrease it. This error propagates to Table II, where every row places Γ (and usually βmax) in the denominator; for example, the qDRIFT row should have Γβmax² in the numerator under the same substitution. The central resource comparison is therefore not established as written.
  2. [Sec. IV, Eq. (58) and proof of Theorem 2] The parameter βmax is defined with dependence on ν, since Eq. (58) contains sqrt(t/(m²ν)) βS and sqrt(t/ν) βEℓ. Because ν itself is set to O(t²m‖O‖Γ/ε), βmax is not independent of Γ, ε, and t, and the substitution into Eq. (63) does not produce a resolved asymptotic bound in fixed problem parameters. Additionally, the proof asserts that βEℓ=1 after Eq. (62) without having imposed this in Sec. IV; if the sub-environment Hamiltonian has a non-unit coefficient, the definition of βmax and the depth estimate change. The theorem should be restated in terms of fixed Hamiltonian parameters with all ν-dependences resolved.
  3. [Sec. IVA, Eq. (68), Eq. (69), and Table II] The general 2k-order Trotter formula in Eq. (68) has (Γβmax²)^{1/2+1/(4k)} in the numerator, which for k=1 gives Γ^{3/4}βmax^{3/2}. However, Eq. (69) and the second-order row of Table II report Γβmax^{3/2} in the denominator. These two expressions cannot both be correct. The comparison between second-order Trotter and SA-LCU in the following paragraph relies on this conflicting pair of formulas and therefore needs to be redone after the scaling is corrected.
minor comments (4)
  1. [Sec. IVB, Fig. 3] The numerical benchmark is not reproducible from the manuscript: no code, raw data, or detailed gate-count tables are provided, and the CNOT counts depend on Qiskit optimization choices and on the Trotter-step selection rules. Please release the code or provide a complete tabulation so the claims can be verified.
  2. [Appendix D, Eq. (A7)] In the proof of Theorem A3, Eq. (A7) writes the second term in the sum as another ΦN(YjXj) term; from Eq. (A6) the second term should be the ΦN(XjYj) composition with XK applied on the left and YK† on the right. This appears to be a typographical error, but it should be fixed for clarity.
  3. [Sec. V, after Eq. (89)] The authors explicitly leave open the finite-K convergence of the non-Markovian collision model to a continuous-time non-Markovian master equation. This limitation should be stated more prominently in the abstract or introduction, since without a convergence bound the non-Markovian portion is a circuit construction rather than an end-to-end simulation with a guaranteed approximation error.
  4. [Sec. IV, paragraph after Eq. (46)] The statement that λ→1/sqrt(Δt) is described as diverging while Δt is small; since Δt=t/ν and ν grows with t and 1/ε, it would be clearer to state the explicit asymptotic regime rather than using the arrow notation, which can be read as a limit in an unspecified parameter.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Lindblad approximation comes from an external bound, SA-LCU is re-derived in the appendix, and resource counts follow from the constructed algorithm rather than from fitted inputs.

full rationale

The paper's claimed derivation chain does not reduce to its own inputs. The collision-model-to-Lindblad step is imported from Ref. [55] (Pocrnic, Segal, Wiebe), which has no author overlap with this paper: Lemma 3 explicitly restates 'Corollary 2.1 of [55]' and is invoked as an external approximation bound. The single-ancilla LCU subroutine is attributed to Refs. [56-58], among which Ref. [58] is a self-citation by author S. Chakraborty, but the paper also says 'We present a derivation of the LCU decomposition in Appendix B for completeness' and provides the truncated-Taylor derivation leading to α ≤ e^{τ²/r} and q = O(log(r/ε)/loglog(r/ε)). Thus the self-citation is not load-bearing. Theorem 1's ζ-analysis is constructive: it chooses r = O(β²Δt²K) to keep the product of LCU weights O(1), and the depth τd = O(β²K²Δt² polylog + KτρE) follows directly from K applications of that chosen circuit. No fitted parameter is renamed as a prediction; the numerical benchmark compares compiled gate counts of the derived formula, and the non-Markovian section explicitly leaves the finite-K convergence scaling open ('we leave the question of the precise scaling of the error in this approximation open'), which is a limitation rather than a circular step. One non-circular correctness concern: Eq. (63) gives τd = O(m³t³βmax²||O||Γ/ε polylog), while Eq. (64) and Table II state Õ(m³t³||O||/(εΓβmax²)), inverting Γ and βmax; this is an algebraic inconsistency in the theorem's statement, not a self-referential reduction, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the SA-LCU decomposition (external, re-derived in App. B), the Ref [55] discretization bound (external, not re-derived), and standard collision-model assumptions (λ=1/√Δt, thermal environment). No free parameters or invented entities are introduced.

assumptions (4)
  • domain assumption The (m,ν)-collision map approximates e^{Lt} in induced 1-norm with error ε when ν ≥ O(t²mΓ/ε) (Lemma 3, Corollary 2.1 of Ref [55]).
    This is the discretization bound that sets the number of collision steps ν. The entire Lindblad complexity in Theorem 2 scales with ν; the bound is cited from Ref [55] and not re-derived.
  • standard math The SA-LCU decomposition of e^{-iτH} with single ancilla and depth Õ(τ²) is valid (Lemma 2, from Refs [56-58], re-derived in Appendix B).
    The algorithm's core subroutine; a published result that is also proved in the appendix.
  • domain assumption The coupling λ = 1/√Δt and the environment thermal state condition Tr_{E_j}[[H_I, ρS⊗ρE]] = 0 are sufficient for the collision model to converge to Lindblad dynamics.
    Standard assumptions in collision model literature (Refs [22,24,55]) to ensure the Born-Markov limit.
  • domain assumption The total Hamiltonian for each collision is a linear combination of Pauli operators with L1 norm β_j.
    Input model for all near-term Hamiltonian simulation methods; standard in the literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Simulating quantum collision models with Hamiltonian simulations using early fault-tolerant quantum computers." pith.science (2026). https://pith.science/paper/WA2BNMG2

@misc{pith2026250421564,
  author       = {Pith},
  title        = {Pith review of: Simulating quantum collision models with Hamiltonian simulations using early fault-tolerant quantum computers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WA2BNMG2}},
  note         = {Machine review of arXiv:2504.21564}
}
abstract

We develop randomized quantum algorithms to simulate quantum collision models, also known as repeated interaction schemes, which provide a rich framework to model various open-system dynamics. The underlying technique involves composing time evolutions of the total (system, bath, and interaction) Hamiltonian and intermittent tracing out of the environment degrees of freedom. This results in a unified framework where any near-term Hamiltonian simulation algorithm can be incorporated to implement an arbitrary number of such collisions on early fault-tolerant quantum computers: we do not assume access to specialized oracles such as block encodings and minimize the number of ancilla qubits needed. In particular, using the correspondence between Lindbladian evolution and completely positive trace-preserving maps arising out of memoryless collisions, we provide an end-to-end quantum algorithm for simulating Lindbladian dynamics. For a system of $n$-qubits, we exhaustively compare the circuit depth needed to estimate the expectation value of an observable with respect to the reduced state of the system after time $t$ while employing different near-term Hamiltonian simulation techniques, requiring at most $n+2$ qubits in all. We compare the CNOT gate counts of the various approaches for estimating the Transverse Field Magnetization of a $10$-qubit XX-Heisenberg spin chain under amplitude damping. Finally, we also develop a framework to efficiently simulate an arbitrary number of memory-retaining collisions, i.e., where environments interact, leading to non-Markovian dynamics. Overall, our methods can leverage quantum collision models for both Markovian and non-Markovian dynamics on early fault-tolerant quantum computers, shedding light on the advantages and limitations of simulating open systems dynamics using this framework.

Figures

Figures reproduced from arXiv: 2504.21564 by the authors.

Figure 1
Figure 1. FIG. 1. The quantum circuit for simulating a [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The circuit to estimate the expectation value of an observable [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. We consider the problem of estimating the average transverse-field magnetization of a [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Non-Markovian evolution via the collision model illustrating the interleaved dynamics between system-environment and the [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Quantum circuit corresponding to each run of Algorithm [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

85 extracted references · 60 canonical work pages

  1. [55]

    Cilluffo, A

    D. Cilluffo, A. Carollo, S. Lorenzo, J. A. Gross, G. M. Palma, andF.Ciccarello,Collisionalpictureofquantumopticswith giant emitters, Phys. Rev. Res.2, 043070 (2020)

    work page 2020

  2. [1]

    Initialize the system and the ancilla in the stateρS and|+⟩, respectively

  3. [2]

    Set ℓ = j(mod m) +1

    For iterations from j = 0 to K− 1, whereK = mν: a. Set ℓ = j(mod m) +1. b. Initialize the environment register in stateρEℓ. c. Draw two i.i.d. samplesX j and Yj from the ensemble Dℓ = Wℓk, αℓk α (ℓ) , where α (ℓ) = ∑k|αℓk| d. Apply the controlled unitaryX (c) j and the anti-controlled unitaryY (a) j to the system. e. Perform a partial trace over the envir...

  4. [3]

    Measure the joint ancilla and system state on the observable (σ x⊗ O) and record the measurement outcome asµi

  5. [4]

    Repeat Steps 1 to 3 a total ofT times

  6. [5]

    Output: Estimated expectation valueµ entangled pure state |ψ⟩ =|00⟩ + e−ω/2|11⟩√ 1 + e−ω , and then tracing out the second qubit

    Compute the final estimateµ as: µ = ζ 2 T T ∑ j=1 µ j, where ζ = ∏K j=1 α ( j). Output: Estimated expectation valueµ entangled pure state |ψ⟩ =|00⟩ + e−ω/2|11⟩√ 1 + e−ω , and then tracing out the second qubit. Henceforth, we will assume that preparingρE j is a constant-depth unitary pro- cedure. We consider that the interaction Hamiltonian corre- sponding...

  7. [6]

    Initialize the system, the first environment register, and the ancilla in statesρS, ρE1, and|+⟩, respectively

    work page

  8. [7]

    Draw two i.i.d

    For iterations from j = 1 to K− 1: a. Draw two i.i.d. samplesX j and Yj from the ensemble D j = Wjℓ, α jℓ α ( j) , where α ( j) = ∑ℓ|α jℓ| b. Apply the controlled unitaryX (c) j and the anti-controlled unitaryY (a) j , controlled on the ancilla qubit with the target being the system register and the first (second) environment register ifj is odd (even). c...

    work page

Show all 85 references
  1. [8]

    samplesXK and YK from the ensemble DK = WKℓ, αKℓ α (K) , where α (K) = ∑ℓ|αKℓ|

    Draw two i.i.d. samplesXK and YK from the ensemble DK = WKℓ, αKℓ α (K) , where α (K) = ∑ℓ|αKℓ|

  2. [9]

    Apply the unitary X (c) K and the anti-controlled unitaryY (a) K , controlled on the ancilla with the target being the system and the first (second) environment register ifK is odd (even)

  3. [10]

    If K is odd, perform a partial trace over the first environment register; otherwise, perform a partial trace over the second environment register

  4. [11]

    Measure the joint ancilla and system state on the observable(σ x⊗ O) and record the measurement outcome asµi

  5. [12]

    Repeat Steps 1 to 7 a total ofT times

  6. [13]

    Compute the final estimateµ as: µ = ζ 2 T T ∑ j=1 µ j, where ζ = ∏K j=1 α ( j). Output: Estimated expectation valueµ Then, the expectation value of O with respect to the state transformed under the approximate map isε-close to the ex- pectation value under the exact map. That ...

    2024

  7. [14]

    Lloyd, Universal quantum simulators, Science273, 1073 (1996)

    S. Lloyd, Universal quantum simulators, Science273, 1073 (1996)

    1996

  8. [15]

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. 21 Somma, Exponential improvement in precision for simulat- ing sparse Hamiltonians, inProceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14 (Association for Computing Machinery, New York...

  9. [16]

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating Hamiltonian dynamics with a trun- cated Taylor series, Phys. Rev. Lett.114, 090502 (2015)

    2015

  10. [17]

    Campbell, Random compiler for fast hamiltonian simu- lation, Phys

    E. Campbell, Random compiler for fast hamiltonian simu- lation, Phys. Rev. Lett.123, 070503 (2019)

    2019

  11. [18]

    G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum3, 163 (2019)

    2019

  12. [19]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The theory of open quantum systems (Oxford University Press, USA, 2002)

    2002

  13. [20]

    Rivas and S

    A. Rivas and S. F. Huelga,Open quantum systems, Vol. 10 (Springer, 2012)

    2012

  14. [21]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Com- pletely Positive Dynamical Semigroups of N Level Systems, J. Math. Phys.17, 821 (1976)

    1976

  15. [22]

    Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics48, 119–130 (1976)

    G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics48, 119–130 (1976)

    1976

  16. [23]

    S. Endo, S. C. Benjamin, and Y. Li, Practical quantum er- ror mitigation for near-future applications, Phys. Rev. X8, 031027 (2018)

    2018

  17. [24]

    Kandala, K

    A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Error mitigation extends the computational reach of a noisy quantum processor, Nature 567, 491 (2019)

    2019

  18. [25]

    Ding, C.-F

    Z. Ding, C.-F. Chen, and L. Lin, Single-ancilla ground state preparation via Lindbladians, Phys. Rev. Res. 6, 033147 (2024)

    2024

  19. [26]

    C.-F. Chen, M. J. Kastoryano, F. G. Brandão, and A. Gilyén, Quantum thermal state preparation, arXiv:2303.18224 10.48550/arXiv.2303.18224 (2023)

    2023 doi arXiv

  20. [27]

    Chakraborty, A

    S. Chakraborty, A. Gilyén, and S. Jeffery, The Power of Block-Encoded Matrix Powers: Improved Regression Tech- niques via Faster Hamiltonian Simulation, in46th Interna- tional Colloquium on Automata, Languages, and Program- ming (ICALP 2019), Leibniz International Proceedings ...

    2019

  21. [28]

    Cleve and C

    R. Cleve and C. Wang, Efficient quantum algorithms for simulating Lindblad evolution, in44th International Collo- quium on Automata, Languages, and Programming (ICALP 2017), Vol. 80 (Schloss Dagstuhl–Leibniz-Zentrum für In- formatik, 2017) pp. 17:1–17:14

    2017

  22. [29]

    A. M. Childs and T. Li, Efficient simulation of sparse Marko- vian quantum dynamics, Quantum Information & Compu- tation 17, 901 (2017)

    2017

  23. [30]

    Li and C

    X. Li and C. Wang, Simulating Markovian open quantum systems using higher-order series expansion, in50th Inter- nationalColloquiumonAutomata, Languages, andProgram- ming (ICALP 2023), Vol. 261 (Schloss Dagstuhl-Leibniz- Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2023) p...

    2023

  24. [31]

    Patel and M

    D. Patel and M. M. Wilde, Wave matrix Lindbladization I: Quantum programs for simulating Markovian dynam- ics, Open Systems & Information Dynamics30, 2350010 (2023)

    2023

  25. [32]

    Patel and M

    D. Patel and M. M. Wilde, Wave matrix Lindbladization II: General Lindbladians, linear combinations, and polynomi- als, Open Systems & Information Dynamics30, 2350014 (2023)

    2023

  26. [33]

    Katabarwa, K

    A. Katabarwa, K. Gratsea, A. Caesura, and P. D. Johnson, Early fault-tolerant quantum computing, PRX Quantum5, 020101 (2024)

    2024

  27. [34]

    Bruneau, A

    L. Bruneau, A. Joye, and M. Merkli, Repeated interactions in open quantum systems, Journal of Mathematical Physics 55, 075204 (2014)

    2014

  28. [35]

    Ciccarello, S

    F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma, Quantum collision models: Open system dynamics from re- peated interactions, Physics Reports954, 1 (2022)

    2022

  29. [36]

    Ciccarello, G

    F. Ciccarello, G. M. Palma, and V. Giovannetti, Collision- model-based approach to non-Markovian quantum dynam- ics, Phys. Rev. A87, 040103 (2013)

    2013

  30. [37]

    Cattaneo, G

    M. Cattaneo, G. De Chiara, S. Maniscalco, R. Zambrini, and G. L. Giorgi, Collision models can efficiently simulate any multipartite Markovian quantum dynamics, Phys. Rev. Lett. 126, 130403 (2021)

    2021

  31. [38]

    P.Strasberg,G.Schaller,T.Brandes,andM.Esposito,Quan- tum and information thermodynamics: A unifying frame- work based on repeated interactions, Physical Review X7, 021003 (2017)

    2017

  32. [39]

    Filipowicz, J

    P. Filipowicz, J. Javanainen, and P. Meystre, Theory of a microscopic maser, Phys. Rev. A34, 3077 (1986)

    1986

  33. [40]

    Barra, Dissipative charging of a quantum battery, Phys

    F. Barra, Dissipative charging of a quantum battery, Phys. Rev. Lett.122, 210601 (2019)

    2019

  34. [41]

    S. Seah, M. Perarnau-Llobet, G. Haack, N. Brunner, and S. Nimmrichter, Quantum speed-up in collisional battery charging, Phys. Rev. Lett.127, 100601 (2021)

    2021

  35. [42]

    Lorenzo, R

    S. Lorenzo, R. McCloskey, F. Ciccarello, M. Paternostro, and G. M. Palma, Landauer’s principle in multipartite open quantum system dynamics, Phys. Rev. Lett.115, 120403 (2015)

    2015

  36. [43]

    Scarani, M

    V. Scarani, M. Ziman, P. Štelmachovič, N. Gisin, and V.Bužek,Thermalizingquantummachines: Dissipationand entanglement, Phys. Rev. Lett.88, 097905 (2002)

    2002

  37. [44]

    A.Manatuly, W.Niedenzu, R.Román-Ancheyta, B.Çakmak, O. E. Müstecaplıoğlu, and G. Kurizki, Collectively enhanced thermalization via multiqubit collisions, Phys. Rev. E99, 042145 (2019)

    2019

  38. [45]

    F. L. S. Rodrigues, G. De Chiara, M. Paternostro, and G.T.Landi,Thermodynamicsofweaklycoherentcollisional models, Phys. Rev. Lett.123, 140601 (2019)

    2019

  39. [46]

    Leitch, N

    H. Leitch, N. Piccione, B. Bellomo, and G. De Chiara, Driven quantum harmonic oscillators: A working medium for ther- mal machines, AVS Quantum Science4, 012001 (2022)

    2022

  40. [47]

    D.Grimmer,E.Brown,A.Kempf,R.B.Mann,andE.Martín- Martínez, Gaussian ancillary bombardment, Phys. Rev. A 97, 052120 (2018)

    2018

  41. [48]

    Hammam, H

    K. Hammam, H. Leitch, Y. Hassouni, and G. De Chiara, Ex- ploitingcoherenceforquantumthermodynamicadvantage, New Journal of Physics24, 113053 (2022)

    2022

  42. [49]

    Ciccarello, Collision models in quantum optics, Quantum Measurements and Quantum Metrology4, 53 (2017)

    F. Ciccarello, Collision models in quantum optics, Quantum Measurements and Quantum Metrology4, 53 (2017)

    2017

  43. [50]

    A. L. Grimsmo, Time-delayed quantum feedback control, Phys. Rev. Lett.115, 060402 (2015)

    2015

  44. [51]

    S. J. Whalen, A. L. Grimsmo, and H. J. Carmichael, Open quantum systems with delayed coherent feedback, Quan- tum Science and Technology2, 044008 (2017). 22

    2017

  45. [52]

    Pichler and P

    H. Pichler and P. Zoller, Photonic circuits with time de- lays and quantum feedback, Phys. Rev. Lett.116, 093601 (2016)

    2016

  46. [53]

    K. A. Fischer, R. Trivedi, V. Ramasesh, I. Siddiqi, and J. Vučković, Scattering into one-dimensional waveguides from a coherently-driven quantum-optical system, Quan- tum 2, 69 (2018)

    2018

  47. [54]

    Fischer, Derivation of the quantum-optical master equa- tion based on coarse-graining of time, Journal of Physics Communications2, 091001 (2018)

    K. Fischer, Derivation of the quantum-optical master equa- tion based on coarse-graining of time, Journal of Physics Communications2, 091001 (2018)

    2018

  48. [56]

    J.A.Gross, C.M.Caves, G.J.Milburn,andJ.Combes,Qubit models of weak continuous measurements: Markovian con- ditional and open-system dynamics, Quantum Science and Technology3, 024005 (2018)

    2018

  49. [57]

    S. Seah, S. Nimmrichter, D. Grimmer, J. P. Santos, V. Scarani, and G. T. Landi, Collisional quantum thermom- etry, Phys. Rev. Lett.123, 180602 (2019)

    2019

  50. [58]

    A. Shu, S. Seah, and V. Scarani, Surpassing the thermal cramér-rao bound with collisional thermometry, Phys. Rev. A 102, 042417 (2020)

    2020

  51. [59]

    Pellegrini and F

    C. Pellegrini and F. Petruccione, Non-Markovian quantum repeated interactions and measurements, J. Phys. A Math. Theor.42, 425304 (2009)

    2009

  52. [60]

    Man, Y.-J

    Z.-X. Man, Y.-J. Xia, and R. Lo Franco, Validity of the Lan- dauer principle and quantum memory effects via collisional models, Phys. Rev. A99, 042106 (2019)

    2019

  53. [61]

    Lorenzo, F

    S. Lorenzo, F. Ciccarello, G. M. Palma, and B. Vacchini, Quantum non-Markovian piecewise dynamics from colli- sion models, Open Syst. Inf. Dyn.24, 1740011 (2017)

    2017

  54. [62]

    Lorenzo, F

    S. Lorenzo, F. Ciccarello, and G. M. Palma, Class of exact memory-kernel master equations, Phys. Rev. A93, 052111 (2016)

    2016

  55. [63]

    D.W.Berry,A.M.Childs,andR.Kothari,Hamiltoniansimu- lationwithnearlyoptimaldependenceonallparameters,in 2015 IEEE 56th Annual Symposium on Foundations of Com- puter Science(2015) pp. 792–809

    2015

  56. [64]

    G. H. Low and I. L. Chuang, Optimal Hamiltonian simu- lation by quantum signal processing, Phys. Rev. Lett.118, 010501 (2017)

    2017

  57. [65]

    A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, To- ward the first quantum simulation with quantum speedup, ProceedingsoftheNationalAcademyofSciences 115,9456 (2018)

    2018

  58. [66]

    A.M.Childs,Y.Su,M.C.Tran,N.Wiebe,andS.Zhu,Theory of Trotter error with commutator scaling, Physical Review X 11, 011020 (2021)

    2021

  59. [67]

    Z.Ding, X.Li,andL.Lin,Simulatingopenquantumsystems using Hamiltonian simulations, PRX Quantum5, 020332 (2024)

    2024

  60. [68]

    Pocrnic, D

    M. Pocrnic, D. Segal, and N. Wiebe, Quantum simula- tion of Lindbladian dynamics via repeated interactions, arXiv:2312.05371 10.48550/arXiv.2312.05371 (2023)

    2023 doi arXiv

  61. [69]

    K. Wan, M. Berta, and E. T. Campbell, Randomized quan- tum algorithm for statistical phase estimation, Phys. Rev. Lett. 129, 030503 (2022)

    2022

  62. [70]

    S. Wang, S. McArdle, and M. Berta, Qubit-efficient random- ized quantum algorithms for linear algebra, PRX Quantum 5, 020324 (2024)

    2024

  63. [71]

    Chakraborty, Implementing any linear combination of unitaries on intermediate-term quantum computers, Quan- tum 8, 1496 (2024)

    S. Chakraborty, Implementing any linear combination of unitaries on intermediate-term quantum computers, Quan- tum 8, 1496 (2024)

    2024

  64. [72]

    Aaronson and P

    S. Aaronson and P. Rall, Quantum approximate counting, simplified, inSymposium on simplicity in algorithms(SIAM,

  65. [73]

    Grinko, J

    D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, Iterative quantum amplitude estimation, npj Quantum Information 7, 52 (2021)

    2021

  66. [74]

    Borras and M

    E. Borras and M. Marvian, Quantum algorithm to simu- late lindblad master equations, Phys. Rev. Res.7, 023076 (2025)

    2025

  67. [75]

    S.Sachdev,QuantumPhaseTransitions ,2nded.(Cambridge University Press, 2011)

    2011

  68. [76]

    J. D. Watson and J. Watkins, Exponentially reduced circuit depths using Trotter error mitigation, arXiv:2408.14385 10.48550/arXiv.2408.14385 (2024)

    2024 doi arXiv

  69. [77]

    J. D. Watson, Randomly compiled quantum simulation with exponentially reduced circuit depths, arXiv preprint arXiv:2411.04240 10.48550/arXiv.2411.04240 (2024)

    2024 doi arXiv

  70. [78]

    Chakraborty, S

    S. Chakraborty, S. Hazra, T. Li, C. Shao, X. Wang, and Y. Zhang, Quantum singular value transforma- tion without block encodings: Near-optimal complexity with minimal ancilla, arXiv preprint arXiv:2504.02385 10.48550/arXiv.2504.02385 (2025)

    2025 doi arXiv

  71. [79]

    J. Kato, K. Wada, K. Ito, and N. Yamamoto, Exponentially accurate open quantum simulation via randomized dissipa- tion with minimal ancilla, arXiv preprint arXiv:2412.19453 10.48550/arXiv.2412.19453 (2024)

    2024 doi

  72. [80]

    Li and C

    X. Li and C. Wang, Succinct description and efficient sim- ulation of non-Markovian open quantum systems, Commu- nications in Mathematical Physics401, 147 (2023)

    2023

  73. [81]

    D. W. Berry, A. M. Childs, Y. Su, X. Wang, and N. Wiebe, Time-dependent Hamiltonian simulation with L1-norm scaling, Quantum4, 254 (2020)

    2020

  74. [82]

    Y.-H. Chen, A. Kalev, and I. Hen, Quantum algorithm for time-dependent hamiltonian simulation by permutation ex- pansion, PRX Quantum2, 030342 (2021)

    2021

  75. [83]

    Watkins, N

    J. Watkins, N. Wiebe, A. Roggero, and D. Lee, Time- dependenthamiltoniansimulationusingdiscrete-clockcon- structions, PRX Quantum5, 040316 (2024)

    2024

  76. [84]

    D.Fang, D.Liu,andR.Sarkar,Time-dependenthamiltonian simulationviamagnusexpansion: Algorithmandsupercon- vergence, Communications in Mathematical Physics406, 1 (2025)

    2025

  77. [85]

    TrEK YK ⃝K−1 j=1 ΦN (Yj X j ) j [ρS] X † K + TrEK YK ⃝K−1 j=1 ΦN (Yj X j ) j [ρS] X † K #  (A7) Then, by the linearity of expectation, we have E[µk] = 1 ζ 2

    M. B. Ruskai, Inequalities for traces on von Neumann al- gebras, Communications in Mathematical Physics26, 280 (1972). 23 Appendix In the Appendix, we provide a comprehensive list of mathematical symbols, an LCU decomposition of unitaries, prove some results concerning distanc...

    1972

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.