pith. sign in

arxiv: 2604.26263 · v2 · submitted 2026-04-29 · 🪐 quant-ph

qSHIFT: An Adaptive Sampling Protocol for Higher-Order Quantum Simulation

Sangjin Lee , Sangkook Choi This is my paper

Pith reviewed 2026-05-07 13:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum simulationadaptive samplinghigher-order error scalingL-independent complexityclassical subroutinenear-term quantum devicesTrotterizationqDRIFT
0
0 comments X

The pith

qSHIFT achieves O(t^{1+r}) error scaling in quantum simulation while keeping gate complexity independent of the number of Hamiltonian terms.

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

The paper introduces qSHIFT as an adaptive sampling protocol for quantum simulation that updates sampling distributions on the fly. Unlike Trotterization, whose circuit depth grows with the number of Hamiltonian terms L, or qDRIFT, whose error scales only as O(t squared), qSHIFT reaches error O(t to the power 1 plus an adjustable r). This is accomplished by solving a classical system of L to the r linear equations in each round to adjust the probabilities. A reader would care because the method promises higher precision without deeper quantum circuits, which could extend what near-term hardware can simulate before decoherence sets in.

Core claim

qSHIFT maintains L-independent gate complexity while achieving an improved error scaling of O(t^{1+r}) for an adjustable parameter r, enabled by a classical subroutine solving L^r linear equations per sampling round.

What carries the argument

Adaptive updating of sampling distributions by solving L^r linear equations classically each round, which cancels lower-order error contributions in the quantum evolution.

If this is right

  • Quantum circuits for simulation can reach higher accuracy at fixed depth and gate count.
  • The protocol's shallower circuits become more compatible with physical error mitigation on noisy hardware.
  • qSHIFT can serve as a high-precision building block for composite algorithms such as qSWIFT or Krylov quantum diagonalization.
  • By raising the parameter r, users can trade classical computation for arbitrarily improved error scaling without touching the quantum resources.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The classical cost of solving L^r equations may limit practical values of r when the number of terms is very large.
  • Hybrid quantum-classical adaptivity of this kind could extend to other tasks such as quantum optimization or open-system dynamics.
  • Pairing qSHIFT with existing error-mitigation techniques might allow simulation of modestly longer times than either method alone.

Load-bearing premise

The adaptive update of sampling distributions via the classical linear solves actually produces the stated O(t^{1+r}) error bound without hidden overheads, instabilities, or additional quantum resources that would re-introduce L dependence.

What would settle it

A numerical test on a small Hamiltonian that measures the observed simulation error as a function of t and checks whether the scaling matches O(t^{1+r}) for chosen r or instead reverts to quadratic scaling, while also verifying that total quantum gate count stays flat as L increases.

Figures

Figures reproduced from arXiv: 2604.26263 by Sangjin Lee, Sangkook Choi.

Figure 1
Figure 1. Figure 1: Numerical demonstrations the (N, r)-qSHIFT and qDRIFT protocols for the 1D transverse field Ising model with (h1, h2) = (1, 0.1) on six qubits. The time-evolution of a physical observable Q = P6 i=1 Zi is estimated for a randomly chosen initial state. Absolute algorithmic errors mitigated by the qSHIFT protocol (green dashed lines) and the qDRIFT protocol (yellow dashed lines) are shown as a function of ev… view at source ↗
read the original abstract

Quantum simulation is a cornerstone application for quantum computing, yet standard methods face a trade-off between circuit depth and accuracy: Trotterization depth scales with the number of Hamiltonian terms $L$, while sampling-based qDRIFT is restricted to $O(t^2)$ error scaling. Here, We introduce qSHIFT, an adaptive sampling protocol that overcomes these limitations. By adaptively updating sampling distributions, qSHIFT maintains $L$-independent gate complexity while achieving an improved error scaling of $O(t^{1+r})$ for an adjustable parameter $r$. This performance is enabled by a classical subroutine solving $L^r$ linear equations per sampling round. Numerical demonstrations confirm the $O(t^{1+r})$ scaling, showcasing qSHIFT as a resource-efficient framework for high-precision quantum simulation. Furthermore, the protocol's reduced circuit depth enhances its compatibility with physical error mitigation, making it a promising candidate for implementation on near-term quantum devices. In addition to its role as a standalone algorithm, qSHIFT can provide a high-precision foundation for modular quantum frameworks such as qSWIFT or Krylov quantum diagonalization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces qSHIFT, an adaptive sampling protocol for quantum Hamiltonian simulation. By using a classical subroutine to solve L^r linear equations per round and adaptively update sampling distributions, the protocol claims to achieve L-independent quantum gate complexity together with an improved error scaling of O(t^{1+r}) for a tunable parameter r. Numerical demonstrations are presented to confirm the scaling, and the method is positioned as compatible with error mitigation and modular frameworks such as qSWIFT or Krylov diagonalization.

Significance. If the adaptive-update mechanism rigorously delivers the stated O(t^{1+r}) bound without introducing hidden L-dependent overheads, precision requirements on the classical solver, or instabilities in the sampling distribution, qSHIFT would represent a useful hybrid improvement over both Trotterization and qDRIFT-style sampling, offering higher accuracy at fixed quantum resources for near-term devices.

major comments (1)
  1. Abstract and error-analysis section: the central claim of an O(t^{1+r}) error bound is asserted on the basis of numerical demonstrations, yet no explicit derivation, variance bound on the updated distribution, or accounting for finite-precision effects in the classical linear solves of L^r equations is supplied; without these, it is impossible to confirm that the adaptive feedback closes the error analysis without reintroducing L dependence or additional sampling overhead.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for a more explicit error analysis. We address the major comment below and have revised the manuscript to incorporate additional analytical details.

read point-by-point responses

  1. Referee: Abstract and error-analysis section: the central claim of an O(t^{1+r}) error bound is asserted on the basis of numerical demonstrations, yet no explicit derivation, variance bound on the updated distribution, or accounting for finite-precision effects in the classical linear solves of L^r equations is supplied; without these, it is impossible to confirm that the adaptive feedback closes the error analysis without reintroducing L dependence or additional sampling overhead.

    Authors: We agree that the original presentation would be strengthened by an explicit derivation. In the revised manuscript we have added a dedicated subsection deriving the O(t^{1+r}) bound. The argument proceeds by expanding the sampling distribution in moments and showing that the classical solution of the L^r linear system enforces exact matching of the first r moments with the ideal distribution; the remainder term then yields an error of order t^{1+r} per segment. We also supply a variance bound for the Monte-Carlo estimator under the updated distribution, confirming that the variance factor remains independent of L. For finite-precision effects we have included a short analysis: the linear system is solved classically with standard double-precision arithmetic; the condition number grows at most polynomially with L for physically relevant Hamiltonians, so any truncation error stays confined to the classical pre-processing step and does not propagate into additional quantum sampling overhead or L-dependent gate counts. These additions close the error analysis without altering the claimed L-independent quantum complexity. revision: yes

Circularity Check

0 steps flagged

No circularity: qSHIFT error bound derived from protocol definition, not reduced to inputs

full rationale

The abstract and context present qSHIFT as a novel adaptive protocol whose O(t^{1+r}) scaling is claimed to follow from the classical linear solves of L^r equations per round, with L-independent quantum complexity maintained by design. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear. The adjustable r is an explicit input to the protocol rather than a post-hoc fit, and numerical confirmation is presented separately from the claimed bound. The derivation chain is self-contained against external benchmarks with no evident reduction of the central claim to its own assumptions by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard quantum simulation error analysis plus one user-chosen parameter r that controls the order; the protocol itself is the main invented component.

free parameters (1)
  • r
    Adjustable positive parameter that sets the target error order; chosen by the user to trade classical cost against quantum error scaling.
axioms (1)
  • standard math Standard quantum mechanics and Trotter/qDRIFT error bounds hold for the underlying Hamiltonian simulation
    The protocol builds on existing quantum simulation theory referenced in the abstract.
invented entities (1)
  • qSHIFT adaptive sampling protocol no independent evidence
    purpose: To achieve L-independent gate complexity with tunable error scaling
    Newly introduced method whose correctness is asserted via the classical subroutine and numerical tests.

pith-pipeline@v0.9.0 · 5484 in / 1433 out tokens · 45776 ms · 2026-05-07T13:38:05.807447+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

52 extracted references · 4 canonical work pages

  1. [1]

    J. I. Cirac and P. Zoller, Goals and opportunities in quan- tum simulation, Nature Physics8, 264 (2012)

    2012

  2. [2]

    Lloyd, Universal quantum simulators

    S. Lloyd, Universal quantum sim- ulators, Science273, 1073 (1996), https://www.science.org/doi/pdf/10.1126/science.273.5278.1073

    work page doi:10.1126/science.273.5278.1073 1996

  3. [3]

    Monroe, W

    C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys.93, 025001 (2021)

    2021

  4. [4]

    Foss-Feig, G

    M. Foss-Feig, G. Pagano, A. C. Potter, and N. Y. Yao, Progress in trapped-ion quantum simulation, Annual Re- view of Condensed Matter Physics16, 145 (2025)

    2025

  5. [5]

    Islam, E

    R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G. D. Lin, L. M. Duan, C. C. Joseph Wang, J. K. Freericks, and C. Monroe, Onset of a quantum phase transition with a trapped ion quantum simulator, Nature Communications2, 377 (2011)

    2011

  6. [6]

    Xiang, K

    Z.-C. Xiang, K. Huang, Y.-R. Zhang, T. Liu, Y.-H. Shi, C.-L. Deng, T. Liu, H. Li, G.-H. Liang, Z.-Y. Mei, H. Yu, G. Xue, Y. Tian, X. Song, Z.-B. Liu, K. Xu, D. Zheng, F. Nori, and H. Fan, Simulating chern insulators on a superconducting quantum processor, Nature Communi- cations14, 5433 (2023)

    2023

  7. [7]

    Viyuela, A

    O. Viyuela, A. Rivas, S. Gasparinetti, A. Wallraff, S. Fil- ipp, and M. A. Martin-Delgado, Observation of topolog- ical uhlmann phases with superconducting qubits, npj Quantum Information4, 10 (2018)

    2018

  8. [8]

    J. Niu, T. Yan, Y. Zhou, Z. Tao, X. Li, W. Liu, L. Zhang, H. Jia, S. Liu, Z. Yan, Y. Chen, and D. Yu, Simulation of higher-order topological phases and related topologi- cal phase transitions in a superconducting qubit, Science Bulletin66, 1168 (2021)

    2021

  9. [9]

    Semeghini, H

    G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Probing topological spin liquids on a programmable quantum simulator, Science374, 1242 (2021), https://www.science.org/doi/pdf/10.1126/science.abi8794

    work page doi:10.1126/science.abi8794 2021

  10. [10]

    Homeier, C

    L. Homeier, C. Schweizer, M. Aidelsburger, A. Fedorov, and F. Grusdt,z 2 lattice gauge theories and kitaev’s toric code: A scheme for analog quantum simulation, Phys. Rev. B104, 085138 (2021)

    2021

  11. [11]

    B. P. Lanyon, C. Hempel, D. Nigg, M. M¨ uller, R. Ger- ritsma, F. Z¨ ahringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, and C. F. Roos, Universal digital quantum sim- ulation with trapped ions, Science334, 57 (2011)

    2011

  12. [12]

    Bauer, S

    B. Bauer, S. Bravyi, M. Motta, and G. K.-L. Chan, Quantum algorithms for quantum chemistry and quan- tum materials science, Chemical Reviews120, 12685 (2020)

    2020

  13. [13]

    Kassal, J

    I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Simulating chemistry using quantum computers, Annual Review of Physical Chem- istry62, 185 (2011)

    2011

  14. [14]

    Reiher, N

    M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Elucidating reaction mechanisms on quantum computers., Proc Natl Acad Sci U S A114, 7555 (2017)

    2017

  15. [15]

    Navickas, R

    T. Navickas, R. J. MacDonell, C. H. Valahu, V. C. Olaya- Agudelo, F. Scuccimarra, M. J. Millican, V. G. Matsos, H. L. Nourse, A. D. Rao, M. J. Biercuk, C. Hempel, I. Kassal, and T. R. Tan, Experimental quantum sim- ulation of chemical dynamics, Journal of the American Chemical Society147, 23566 (2025). 6 Protocol Gate complexity Sampling complexity 1st or...

    2025

  16. [16]

    Malpathak, S

    S. Malpathak, S. D. Kallullathil, I. Loaiza, S. Fomichev, J. M. Arrazola, and A. F. Izmaylov, Trotter simulation of vibrational hamiltonians on a quantum computer, Jour- nal of Chemical Theory and Computation22, 95 (2026)

    2026

  17. [17]

    T. N. Georges, M. Bothe, C. S¨ underhauf, B. K. Berntson, R. Izs´ ak, and A. V. Ivanov, Quantum simulations of chemistry in first quantization with any basis set, npj Quantum Information11, 55 (2025)

    2025

  18. [18]

    N. W. Andrew M. Childs, Hamiltonian simulation using linear combinations of unitary operations, Quantum In- formation and Computation12, 0901 (2012)

    2012

  19. [19]

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Simulating hamiltonian dynamics with a truncated taylor series, Phys. Rev. Lett.114, 090502 (2015)

    2015

  20. [20]

    G. H. Low and I. L. Chuang, Optimal hamiltonian sim- ulation by quantum signal processing, Phys. Rev. Lett. 118, 010501 (2017)

    2017

  21. [21]

    D. W. Berry, A. M. Childs, A. Ostrander, and G. Wang, Quantum algorithm for linear differential equations with exponentially improved dependence on precision, Com- munications in Mathematical Physics356, 1057 (2017)

    2017

  22. [22]

    A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A. Spielman, Exponen- tial algorithmic speedup by a quantum walk, in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC ’03 (Association for Computing Machinery, New York, NY, USA, 2003) pp. 59–68

    2003

  23. [23]

    A. M. Childs, On the relationship between continuous- and discrete-time quantum walk, Communications in Mathematical Physics294, 581 (2010)

    2010

  24. [24]

    A. M. C. Dominic W. Berry, Black-box hamiltonian sim- ulation and unitary implementation, Quantum Informa- tion and Computation12, 0029 (2012)

    2012

  25. [25]

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

    2019

  26. [26]

    M. Suzuki, General theory of fractal path inte- grals with applications to many-body theories and statistical physics, Journal of Mathematical Physics 32, 400 (1991), https://pubs.aip.org/aip/jmp/article- pdf/32/2/400/19166143/400 1 online.pdf

    1991

  27. [27]

    Hatano and M

    N. Hatano and M. Suzuki, Finding expo- nential product formulas of higher orders, in Quantum Annealing and Other Optimization Methods, edited by A. Das and B. K. Chakrabarti (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005) pp. 37–68

    2005

  28. [28]

    A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Theory of trotter error with commutator scaling, Phys. Rev. X11, 011020 (2021)

    2021

  29. [29]

    G. C. Knee and W. J. Munro, Optimal trotterization in universal quantum simulators under faulty control, Phys. Rev. A91, 052327 (2015)

    2015

  30. [30]

    S. A. Chin, Explicit symplectic integrators for solving non-separable hamiltonians (2009), arXiv:physics/0608043 [physics.comp-ph]

    work page arXiv 2009

  31. [31]

    Chen, H.-Y

    C.-F. Chen, H.-Y. Huang, R. Kueng, and J. A. Tropp, Concentration for random product formulas, PRX Quan- tum2, 040305 (2021)

    2021

  32. [32]

    P. K. Faehrmann, M. Steudtner, R. Kueng, M. Kieferova, and J. Eisert, Randomizing multi-product formulas for Hamiltonian simulation, Quantum6, 806 (2022)

    2022

  33. [33]

    R. D. Ruth, A canonical integration technique, IEEE Transactions on Nuclear Science30, 2669 (1983)

    1983

  34. [34]

    Forest and R

    E. Forest and R. D. Ruth, Fourth-order symplectic in- tegration, Physica D: Nonlinear Phenomena43, 105 (1990)

    1990

  35. [35]

    Layden, First-order trotter error from a second-order perspective, Phys

    D. Layden, First-order trotter error from a second-order perspective, Phys. Rev. Lett.128, 210501 (2022)

    2022

  36. [36]

    Campbell, Random compiler for fast hamiltonian sim- ulation, Phys

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

    2019

  37. [37]

    Nakaji, M

    K. Nakaji, M. Bagherimehrab, and A. Aspuru-Guzik, High-order randomized compiler for hamiltonian simu- lation, PRX Quantum5, 020330 (2024)

    2024

  38. [38]

    Piccinelli, A

    S. Piccinelli, A. Baiardi, S. Barison, M. Rossmannek, A. C. Vazquez, F. Tacchino, S. Mensa, E. Altamura, A. Alavi, M. Motta, J. Robledo-Moreno, W. Kirby, K. Sharma, A. Mezzacapo, and I. Tavernelli, Quan- tum chemistry with provable convergence via randomized sample-based krylov quantum diagonalization (2026), arXiv:2508.02578 [quant-ph]

    work page arXiv 2026

  39. [39]

    Temme, S

    K. Temme, S. Bravyi, and J. M. Gambetta, Error mitiga- tion for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017)

    2017

  40. [40]

    van den Berg, Z

    E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse pauli–lindblad models on noisy quantum processors, Na- 7 Algorithm 1 qSHIFT Input:N,V={V i=1,···,L =e −i tλ N },p,V S =1,r, initial state|ψ⟩ forp= 1,· · ·, N/rdo SolveL r variable linear system and obtainp ⃗ s. X ⃗ s p⃗ s∂q t (VSV⃗ s)† QVSV⃗ s t=0 =∂ q t e...

    2023

  41. [41]

    Li and S

    Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X7, 021050 (2017)

    2017

  42. [42]

    Y. Kim, C. J. Wood, T. J. Yoder, S. T. Merkel, J. M. Gambetta, K. Temme, and A. Kandala, Scalable error mitigation for noisy quantum circuits produces competi- tive expectation values, Nature Physics19, 752 (2023)

    2023

  43. [43]

    A. M. Childs and R. Kothari, Limitations on the simu- lation of non-sparse hamiltonians, Quantum Information and Computation10, pp 0669 (2010)

    2010

  44. [44]

    D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, Ef- ficient quantum algorithms for simulating sparse hamil- tonians, Communications in Mathematical Physics270, 359 (2007). 8 Appendix A: Details of qDRIFT calculations

    2007

  45. [45]

    We consider a Hamiltonian, H= LX i=1 hiHi,(∀ ihi >0), and a physical observableQthat we want to simulate with respect to an initial state|ψ⟩

    Mechanism of the qDRIFT In this section, we review the original qDRIFT algorithms. We consider a Hamiltonian, H= LX i=1 hiHi,(∀ ihi >0), and a physical observableQthat we want to simulate with respect to an initial state|ψ⟩. The purpose of quantum simulation is to estimate ⟨Q(t)⟩ideal =⟨ψ|e itH Qe−itH |ψ⟩, as precisely as possible using quantum circuits. ...

  46. [46]

    V ariance of measurement outcomes from qDRIFT In this section, we compute the variance of the qDRIFT protocol to estimate sampling complexity. The variance of qDRIFT is defined as var(⟨Q⟩qDRIF T ) = LX s1,···sN=1 p⃗ s⟨ψ|V † ⃗ sQV⃗ s|ψ⟩2 − LX s1,···sN=1 p⃗ s⟨ψ|V † ⃗ sQV⃗ s|ψ⟩ !2 wherep ⃗ s=p s1 · · ·p sN andV ⃗ s=V s1 · · ·V sN . The first term is LX s1,··...

  47. [47]

    To determinep ⃗ sfor sampling,P ⃗ sp⃗ sV † ⃗ sQV⃗ stoe it r N H Qe−it r N H from lowest order

    Mechanism of algorithm In this section, we provide detailed calculations for the main classical subroutine in qSHIFT. To determinep ⃗ sfor sampling,P ⃗ sp⃗ sV † ⃗ sQV⃗ stoe it r N H Qe−it r N H from lowest order. 11 We expandP ⃗ sp⃗ sV † ⃗ sQV⃗ s, X ⃗ s p⃗ sV † ⃗ sQV⃗ s = X ⃗ s ∞X n1,···,n r=0 p⃗ s itλ N n1+···nr [Hsr ,· · ·[H s3 ,[H s2 ,[H s1 ,Q] n1]n1 ·...

  48. [48]

    V ariance of measurement outcomes from qSHIFT In this section, let us compute the variance qSHIFT. Since the ensemble average of qSHIFT is written as ⟨Q(t)⟩qSHIF T =P ⃗ sq⃗ s Z(p⃗ s)sign(p⃗ s)⟨V † ⃗ sQV⃗ s⟩ , the variance is defined as follows, var⟨Q⟩qSHIF T = X ⃗ s q⃗ s Z(p⃗ s)sign(p⃗ s)⟨V † ⃗ sQV⃗ s⟩ 2 − X ⃗ s q⃗ s Z(p⃗ s)sign(p⃗ s)⟨V † ⃗ sQV⃗ s⟩ !2 . T...

  49. [49]

    Following our protocol, the drawing set is V={V 1 =e −i tλ 2 H1 , V2 =e −i tλ 2 H2 }

    (L= 2,N= 2,r= 1) Consider H=h 1H1 +h 2H2. Following our protocol, the drawing set is V={V 1 =e −i tλ 2 H1 , V2 =e −i tλ 2 H2 }. At the first round, we drawV i with probabilityp i = hi λ . Next, for the drawn operatorV i, we determine the probability for the next drawing. To set the probability, we compare two operators, 2X j=1 p(i) j V † j V † i QViVj, = ...

  50. [50]

    (L= 2,N= 2,r= 2) Next, under the same setting, we consider more drawing at once. Compare X ij pijV † j V † i QViVj, = X ij pijQ, + X ij pij itλ 2 ([Hi,Q] + [Hj,Q]), + X ij pij 1 2 itλ 2 2 ([Hi,Q] 2 + [Hj,Q] 2 + 2[Hj,[H i,Q]]), +O(t 3) to eitH Qe−itH , =Q+ 2X i=1 (it) [hiHi,Q] + 2X i=1 1 2 (it)2 [hjHj,[h iHi,Q]] +O(t 3), which gives X ij pij = 1, λp 11 + λ...

  51. [51]

    We draw the first operatorV i with probabilityp i

    (L= 2,N= 3,r= 2) As another example, we consider (L= 2, N= 3, r= 2) case. We draw the first operatorV i with probabilityp i. To determine the probability of drawing the next two operators, consider X a,b p(i) ab V † abV † i QViVab, = X a,b p(i) ab Q + X a,b p(i) ab itλ 3 ([Hi,Q] + [Ha,Q] + [Hb,Q]) + X a,b p(i) ab 1 2 itλ 3 2 ([Hi,Q] 2 + [Ha,Q] 2 + [Hb,Q] ...

  52. [52]

    (L= 2,N= 3,r= 3) As the last example, we provide the probability distribution following (N= 3, r= 3)-qSHIFT protocol with details of calculations. X ijk pijk(ViVjVk)†QViVjVk, = X ijk pijk Q + X ijk pijk itλ 3 ([Hi,Q] + [Hj,Q] + [Hk,Q]) + X ijk pijk itλ 3 2 1 2! ([Hi,Q] 2 + [Hj,Q] 2 + [Hk,Q] 2) + [Hi,[H j,Q]] + [Hj,[H k,Q]] + [Hi,[H k,Q]] + X ijk pijk itλ ...