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REVIEW 3 major objections 5 minor 151 references

Euclidean-Monte-Carlo-informed ground-state preparation for quantum simulation of scalar field theory

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Classical Monte-Carlo data alone can determine a polynomial-size quantum circuit that prepares the phi-4 ground state.

desk verdict The pipeline is real and the circuit translation is strong, but the multimode claim of closeness to the ground state is only supported by energy and targeted moments, not by any overlap estimate. read the letter →

arxiv 2506.02313 v1 pith:VJ7WTGKK submitted 2025-06-02 quant-ph hep-lathep-phnucl-th

Navya Gupta , Christopher David White , Zohreh Davoudi This is my paper
classification quant-phhep-lathep-phnucl-th
keywords ground-statepreparationquantumsimulationscalarfieldtheoryphi^4Euclideanpath-integralMonteCarlostellarhierarchymomentoptimizationcircuits
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 claims that a classical computer can fully determine an efficient quantum circuit that prepares a state close to the ground state of an interacting scalar field theory, without any quantum-classical feedback loop. The demonstration target is the $\phi^4$ theory in 1+1 dimensions on a lattice of $N=10$ sites, and the Euclidean Monte-Carlo information enters through selected ground-state correlation functions ("moments") used as penalties in a variational optimization. The proposed pipeline has three parts: a bosonic ansatz built from the stellar hierarchy, a moment-optimization loss that adds squared deviations of target moments to the energy, and a translation of the optimized ansatz into a qubit circuit whose gate count is polynomial in system size. At tested couplings the moment-optimized states keep the ground-state energy nearly as low as energy-only optimization while improving the reproduction of non-Gaussian moment ratios and, separately, non-local two-point functions. If the ansatz is expressive enough, the practical consequence is that expensive ground-state preparation can be shifted from quantum hardware to a classical computation guided by Monte-Carlo data.

What carries the argument

The load-bearing object is the $(R,Q)$ ansatz of the stellar hierarchy: a tensor product of identical single-mode squeezing operators acting on a sparse, symmetric polynomial core state of bounded rank $R$ and span $Q$. Its usefulness is that the core has support on only $N|c_{R,Q}|$ Fock states, so all required expectation values are classically computable in time independent of system size, while the state still carries enough non-Gaussian structure to bend moment ratios away from Gaussian values. The companion mechanism is the moment-optimization loss function, which adds squared fractional deviations of selected PIMC-sourced ground-state moments to the energy; the weights $w_{\hat O}$ interpolate between a pure energy-minimized state and a state tuned to reproduce chosen correlations.

What would settle it

Compute, for $N=10$ or smaller at couplings $(m^2,\lambda)=(0.6,1.5)$ through $(0.1,0.25)$, a high-precision numerical ground state and evaluate the fidelity $|\langle\Omega|\psi\rangle_{R,Q}|^2$ for both the energy-optimized and moment-optimized parameters. If the moment-optimized fidelity is substantially worse than the fidelity implied by the energy discrepancy, or if adding target moments moves the state away from the exact ground state while improving moment matching, then the central assumption fails.

Watch

Extended reading notes

Core claim

The central discovery is a workflow that turns Euclidean correlation data into a fixed, classically known quantum circuit. The paper proposes the $(R,Q)$ ansatz, $|\psi\rangle_{R,Q}=\left[\otimes_{j=0}^{N-1}\hat S_j(r)\right] C_{R,Q}(\hat a^\dagger)|0\rangle$, where $\hat S_j(r)$ is a single-mode squeezer and $C_{R,Q}$ is a polynomial of creation operators whose terms are symmetric under parity, time reversal, lattice translation, and lattice inversion, with rank $R$ (total boson number) and span $Q$ (support within neighboring sites). Because the core state has only $N|c_{R,Q}|$ nonzero Fock components, expectation values of local and two-point operators can be evaluated classically in $O(|c_{R,Q}|^2Q^2)$ time independent of $N$. The paper's moment-optimization procedure minimizes the energy plus a weighted sum of squared deviations of selected moments from their Euclidean path-integral Monte-Carlo values; it demonstrates that for a $(2,2)$ ansatz this can pull the local moment ratios $R_{2n}=\langle\hat\phi^{2n}\rangle/\left[(2n-1)!!\langle\hat\phi^2\rangle^n\right]$ toward the non-Gaussian Monte-Carlo values and, in a separate optimization, improve $\langle\hat\phi_0\hat\phi_4\rangle$, at an energy cost that is small compared with the spectral gap for most tested couplings. The same structure is then translated to a qubit circuit: the core state is prepared by a sparse state-preparation algorithm using $O(N^2R|c_{R,Q}|)$ CNOT gates, and the squeezing operators are Trotterized on a truncated Fock space with cutoffs chosen to meet a target fidelity.

Load-bearing premise

The argument rests on the assumption that the $(R,Q)$ ansatz, identical single-mode squeezers applied to a local core of rank $R$ and span $Q$, is flexible enough at the tested couplings that matching the energy and a few Monte-Carlo moments forces the state close to the true ground state; multimode fidelity is not reported to verify this.

Editorial extensions

If this is right

  • For the tested couplings, a classical side computation can produce a complete quantum circuit for the $\phi^4$ ground state that requires no variational feedback from quantum hardware.
  • Moment optimization selects among low-energy ansatz states: it can improve non-Gaussian moment ratios or non-local correlations that energy minimization alone leaves wrong, at energy penalties below the spectral gap except near the continuum limit.
  • The efficiency statements carry over to any theory whose ground state is well approximated by an $(R,Q)$-type core: the circuit depth is polynomial in $N$, $R$, $Q$, and the boson cutoff $\Lambda$, with classical precomputation cost $O(N^3 R |c_{R,Q}|^2 \log(N|c_{R,Q}|))$.
  • Since the PIMC moments carry statistical errors and the pipeline propagates them to optimized parameters, the output is naturally an ensemble of circuits; repeated runs on quantum hardware can turn this into an uncertainty on simulated real-time quantities.

Reading between the lines

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

  • One implication the authors do not develop is that the method gives a concrete way to resolve cases where many candidate states have nearly the same energy: PIMC moments pick the one whose correlation structure matches the true state, suggesting the loss function could be tuned to whatever observable a later dynamics simulation needs.
  • A direct stress test would compute the actual multimode fidelity against a high-precision numerical ground state at small $N$; the paper only verifies energy and moments, so this would reveal whether moment-matching is sufficient to guarantee closeness in state space.
  • Near the continuum limit the correlation length diverges, so $R$ and $Q$ likely have to grow with $N$; then the classical cost and circuit depth cease to be independent of system size, and the pipeline would need either a better ansatz or quantum-assisted moment evaluation to remain efficient.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript develops and tests a classical pipeline for preparing approximate ground states of (1+1)-dimensional lattice phi^4 theory on a quantum simulator. The three elements are: (i) a finite-stellar-rank bosonic ansatz, specialized to a tensor product of single-mode squeezers applied to a rank-R, span-Q polynomial core state; (ii) a moment-optimization loss that augments energy minimization with penalties for deviations from PIMC-computed ground-state moments; and (iii) a translation of the optimized ansatz into qubit circuits using sparse state preparation for the core and truncated, Trotterized squeezing operators for the Gaussian part. The paper validates the ansatz in (0+1)D with explicit fidelities and energy discrepancies, presents multimode N=10 results for energies, two-point functions, local moments, and moment ratios, and provides resource estimates with error bounds for the circuit implementation.

Significance. If the central claim is established, the paper offers a valuable and timely contribution: a classical route from Euclidean Monte-Carlo data to a fixed, polynomial-size quantum circuit for a non-trivial interacting field-theory ground state, without a quantum feedback loop. The single-mode calculations are clean and the fidelity/energy tables provide genuine support for the ansatz in that sector. The circuit-translation section is also a strength: it gives concrete gate counts, truncation bounds, and Trotter-error bounds, and it is honest about the looseness of those bounds. The main weakness is that the multimode evidence does not certify that the optimized states are close to the true ground state: no fidelity, no delta-E, and no overlap with an independent ground-state calculation is reported for N=10. Energy proximity is shown in the paper itself to be insufficient to distinguish very different ansatz states, so the central claim remains contingent on additional evidence.

major comments (3)
  1. [Sec. IV B, Eq. (3), Figs. 7-14] The central claim that the pipeline prepares states close to the (1+1)D ground state is not supported by the multimode numerical evidence. The only quantitative closeness metric reported for the multimode case is the Hamiltonian expectation value (Fig. 7); no fidelity, no delta-E, and no overlap with an independently obtained ground state is reported. Equation (3) provides a lower bound F >= 1 - delta-E, but delta-E is never evaluated for N=10, even though the PIMC data include an excited-state band (top panels of Fig. 7) from which the gap and hence delta-E could be estimated. Since Fig. 7 shows that the GEP and the (R,Q) ansatze attain comparable energies while differing substantially in two-point functions and moment ratios, energy proximity is demonstrably insufficient to certify closeness; Appendix A makes the same point analytically. I request either direct fidelities against an exact or exact-diagonalization ground state in the truncated Fock space for N=10, or at minimum a table of delta-E for the (R,Q) ansatze at the four couplings, with the PIMC gap used as the denominator.
  2. [Sec. IV A, Eqs. (25)-(28), Fig. 7] The restricted (R,Q) ansatz has a structural limitation that interacts with the closeness claim: with passive rotations dropped and span truncation Q, all cross-mode correlations must be carried by the rank-R, span-Q core, and <phi_0 phi_j> is identically zero for j > 2Q (stated in the text following Fig. 7). At the tested couplings, PIMC sees small but nonzero <phi_0 phi_5> (Figs. 20-23), which the (R,Q) ansatz cannot represent at Q=2. This does not invalidate the variational method, but it means that matching energy and the low-j two-point functions does not constrain the unconstrained tail of the correlation function. Without additional information such as fidelity, delta-E, or an un-targeted observable sensitive to the tail, the ansatz could match the plotted moments yet remain far from the true ground state. Please quantify the contribution of the j>2Q tail, or otherwise bound the neglected correlations, in assessing the multimode results.
  3. [Sec. IV B, Eq. (4), Figs. 9-12] The moment-optimization demonstrations are fits rather than independent tests: the loss function in Eq. (4) directly penalizes discrepancies in the target moments, so the monotone decrease in the target-moment discrepancy in Figs. 9 and 12 is guaranteed by construction and cannot be read as evidence that PIMC moments are predictive or that the ansatz is closer to the true ground state. The meaningful validation would be improvement in non-target moments or in quantities absent from the loss, or a demonstrated relation between moment matching and fidelity. As presented, the paper's stronger claims rest on the energy and qualitative correlation plots, which brings the assessment back to Major Comment 1. I recommend that the authors state this distinction explicitly and use a non-target observable such as a connected four-point function or the tail two-point function as a validation metric.
minor comments (5)
  1. [Abstract and Sec. VI] The sentence 'The resulting states yield comparable ground-state energy estimates but exhibit distinct correlations and local non-Gaussianity' is vague; it should specify distinct from which reference states and for which couplings this statement holds.
  2. [Sec. V A, Eq. (42)] There is a typo in 'we we have assumed' near Eq. (42); please correct.
  3. [Sec. V C, Eq. (53) and Table II] The resource estimates in Table II are derived from deliberately loose analytical bounds, as the authors note; it would be helpful to state explicitly that the reported Lambda and K are upper-resource estimates rather than the minimal resources found numerically.
  4. [Fig. 7] The log-scale insets for the two-point functions make it difficult to see that some ansatz values are exactly zero; use a marker or annotation to indicate the zero values rather than points that appear to be missing from the plot.
  5. [Appendix B, Eq. (B7)] The virial theorem estimate for <pi^2> relies on the specific form of the potential; please state the assumptions explicitly so that the PIMC energy estimate is fully reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: PIMC moments are explicit external inputs; target-moment reductions are the optimization objective, not hidden predictions.

full rationale

The paper's derivation chain is a variational construction rather than a prediction from first principles, and its load-bearing claims do not reduce to their inputs by definition. PIMC ground-state moments enter the loss function in Eq. (4) as external inputs; minimizing that loss by construction reduces the squared deviations of the target moments, so the monotone decreases in Figs. 9 and 12 are properties of the optimization objective, not independent predictions. The paper presents them as optimization outcomes and does not use them to validate the method. The central claims that do carry weight—comparable ground-state energy, distinct correlation and non-Gaussianity structure for a fixed ansatz, and polynomial circuit cost—rest on independent inputs: energy minimization is performed directly against the Hamiltonian, the GEP comparison provides an external baseline, and the circuit translation uses standard sparse-state-preparation and Trotter-error techniques. Self-citations (e.g., Refs. [74, 75, 91, 118]) appear as reviews or as a squeezing-implementation gadget and are not used to forbid alternatives or to justify the ansatz itself. The stellar-hierarchy ansatz is attributed to external references [93, 94], not to the authors' prior work. The paper also explicitly acknowledges the multimode evidentiary limitation in Sec. II: 'we will compute both energy discrepancy and fidelity for the single-mode, i.e., the (0+1)D case, but restrict the analysis to just energy discrepancy for the multimode.' That is a stated expressiveness and verification caveat, not a circular step. No equation is equivalent to its own input by construction beyond the explicit definition of the moment-optimization loss function in Eq. (4).

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new particles or mediators are introduced. The central claim rests on the expressibility of the (R,Q) ansatz and the reliability of PIMC-sourced moments as optimization targets; all fit parameters are explicitly labeled, and the main hidden assumptions concern ansatz expressibility and Monte Carlo faithfulness.

free parameters (5)
  • Moment-optimization weights w = 12.5, 25, 50, 100, 200, 400 for moment-ratio optimization; 1.25e4 to 5e5 for two-point optimization
    Chosen by experimentation in Sec. IV B, not derived; they control the trade-off between energy and moment matching.
  • Ansatz variational parameters (r, A, B0, B1, B2, C coefficients) = Optimized per (m^2, lambda), e.g., r values in Fig. 8 and Table II
    These are the optimized degrees of freedom of the ansatz, not externally fixed constants.
  • Target moment set T = T = {phi^6_j, phi^8_j, phi^10_j} or T = {phi_0 phi_4}
    Selected by physical intuition; the choice affects which observables the prepared state reproduces.
  • Ansatz ranks R and spans Q = (2,1), (2,2), (4,1), (4,2)
    Truncations chosen for numerical tractability; the accuracy of the approximation depends on them.
  • PIMC temporal parameters T and theta = T = 10, theta in {0.4, 0.2, 0.1}
    Numerical choices for the PIMC extrapolation to zero temporal lattice spacing, reported in Appendix B.
assumptions (6)
  • domain assumption The ground state is non-degenerate with a non-zero spectral gap, so Eq. (3) gives a fidelity lower bound from energy discrepancy.
    Assumed at the start of Sec. II; needed for the variational justification and for interpreting delta_E as a fidelity proxy.
  • domain assumption The ground state of the phi^4 Hamiltonian has infinite stellar rank, so finite-rank ansatzes can only approximate it.
    Stated in Sec. III A and Sec. IV A, citing Ref. [95]. The approximation error is the entire point of the ansatz; if the ground state had small finite rank, the problem would be trivial.
  • domain assumption PIMC-computed Euclidean correlation functions are faithful proxies for ground-state expectation values of the Minkowski lattice Hamiltonian after T to infinity and theta to 0 extrapolations.
    Used throughout Sec. IV B; relies on the absence of a sign problem and on the validity of the extrapolations described in Appendix B.
  • ad hoc to paper The Q-truncated core polynomial can capture the relevant non-local correlations for N = 10 at the tested couplings.
    Eqs. (25) and (26) impose a span-Q truncation beyond the stellar-rank structure; the authors acknowledge that two-point functions vanish for j > 2Q.
  • standard math The sparse state preparation algorithm of Ref. [110] correctly prepares the core state with the quoted gate counts.
    Invoked in Sec. V B; it is a published algorithm, not re-proved in this paper.
  • domain assumption The truncation and Trotter error bounds in Appendix E are valid and sufficiently tight to choose Lambda and K.
    Table II relies on these bounds; the authors note the bounds are conservative and state-independent.

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Cite this review

Pith. "Pith review of Euclidean-Monte-Carlo-informed ground-state preparation for quantum simulation of scalar field theory." pith.science (2026). https://pith.science/paper/VJ7WTGKK

@misc{pith2026250602313,
  author       = {Pith},
  title        = {Pith review of: Euclidean-Monte-Carlo-informed ground-state preparation for quantum simulation of scalar field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJ7WTGKK}},
  note         = {Machine review of arXiv:2506.02313}
}
read the original abstract

Quantum simulators offer great potential for investigating dynamical properties of quantum field theories. However, preparing accurate non-trivial initial states for these simulations is challenging. Classical Euclidean-time Monte-Carlo methods provide a wealth of information about states of interest to quantum simulations. Thus, it is desirable to facilitate state preparation on quantum simulators using this information. To this end, we present a fully classical pipeline for generating efficient quantum circuits for preparing the ground state of an interacting scalar field theory in 1+1 dimensions. The first element of this pipeline is a variational ansatz family based on the stellar hierarchy for bosonic quantum systems. The second element of this pipeline is the classical moment-optimization procedure that augments the standard variational energy minimization by penalizing deviations in selected sets of ground-state correlation functions (i.e., moments). The values of ground-state moments are sourced from classical Euclidean methods. The resulting states yield comparable ground-state energy estimates but exhibit distinct correlations and local non-Gaussianity. The third element of this pipeline is translating the moment-optimized ansatz into an efficient quantum circuit with an asymptotic cost that is polynomial in system size. This work opens the way to systematically applying classically obtained knowledge of states to prepare accurate initial states in quantum field theories of interest in nature.

Figures

Figures reproduced from arXiv: 2506.02313 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Minimum-energy wavefunctions for the (0+1)D [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (28 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Percent errors in the moments [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Fidelity [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Minimum Hamiltonian expectation values (top panels), two-point functions (middle panels), and local [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Optimized value of the squeezing parameter [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Sum of squared discrepancies in the target moments [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Optimization of the moment ratio for the ( [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Optimized values of the ( [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Sum of squared discrepancies in the target moments [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Optimization of the two-point function for the ( [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: displays the values of the optimized pa￾rameters as a function of the weight w. The largest shifts and errors are associated with the parameter B2 (which accompanies the ϕˆ jϕˆ j+2 term in the core-state polynomial). This is perhaps not surprising because this paramet…
Figure 15
Figure 15. Figure 15: FIG. 15. Behavior of the minimum-energy and maximum-fidelity rank-2 ansatz for the ground state of the squeezed Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Extrapolations of the one- and two-point functions of [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Extrapolations of the one- and two-point functions of [PITH_FULL_IMAGE:figures/full_fig_p027_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Extrapolations of the one- and two-point functions of [PITH_FULL_IMAGE:figures/full_fig_p028_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Extrapolations of the one- and two-point functions of [PITH_FULL_IMAGE:figures/full_fig_p028_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Extrapolations of the two-point correlation functions to zero temporal lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p029_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Extrapolations of the two-point correlation functions to zero temporal lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p029_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Extrapolations of the two-point correlation functions to zero temporal lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p030_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. Extrapolations of the two-point correlation functions to zero temporal lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p030_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Extrapolations of the moment ratios to zero temporal lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p031_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Extrapolations of the moment ratios to zero temporal lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p031_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Extrapolations of the moment ratios to zero temporal lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p032_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Extrapolations of the moment ratios to zero temporal lattice spacing [PITH_FULL_IMAGE:figures/full_fig_p032_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. Effective-mass plots corresponding to ( [PITH_FULL_IMAGE:figures/full_fig_p033_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Effective-mass plots corresponding to ( [PITH_FULL_IMAGE:figures/full_fig_p033_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30. Effective-mass plots corresponding to ( [PITH_FULL_IMAGE:figures/full_fig_p034_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31. Effective-mass plots corresponding to ( [PITH_FULL_IMAGE:figures/full_fig_p034_31.png]
Figure 32
Figure 32. Figure 32: FIG. 32. Boson-number distribution for squeezed Fock states. Top panels show [PITH_FULL_IMAGE:figures/full_fig_p040_32.png]

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Works this paper leans on

151 extracted references · 45 canonical work pages

  1. [1]

    Initialization: Initialize T = S = [0000, 0101, 0110, 1011, 1100, 1101], dif qubits = [], dif vals = []

  2. [2]

    , n} and look for the first qubit b such that the sizes of the sets T0 := {x ∈ T |x[b] = 0 } and T1 := {x ∈ T |x[b] = 1 } are as unequal as possible (with neither set being empty)

    The first WHILE loop: Scan over all qubits b ∈ {1, . . . , n} and look for the first qubit b such that the sizes of the sets T0 := {x ∈ T |x[b] = 0 } and T1 := {x ∈ T |x[b] = 1 } are as unequal as possible (with neither set being empty). This turns out to be b = 1; in this case, T0 = [0000, 1011] and T1 = [0101, 0110, 1100, 1101], i.e., |T0| = 2 and |T1| ...

  3. [3]

    Also remove the last value of dif vals

    Preparation for the second WHILE loop: Remove the last value appended to dif qubits, i.e., b = 0; store it as dif = 0. Also remove the last value of dif vals. Finally, store the single element of T as x1. This leaves x1 = 1011, dif = 0, dif qubits = [1], dif vals = [0]. Now, define a new set T ′ which consists of all those strings x ∈ S which take values ...

  4. [4]

    Since |T ′| is already 1, the loop does not run in this case

    The second WHILE loop: The second WHILE loop runs in exactly the same way as the first WHILE loop, except that the initial conditions now involve the set T ′, and the updated values of dif qubits and dif vals defined above. Since |T ′| is already 1, the loop does not run in this case

  5. [5]

    Thus, the relevant variable values at this stage are x1 = 1011, x2 = 0000, dif = 0, dif qubits = [1], dif vals = [0]

    Preparation for circuit building: Assign the single element in T ′ to the variable x2. Thus, the relevant variable values at this stage are x1 = 1011, x2 = 0000, dif = 0, dif qubits = [1], dif vals = [0]. To summarize, the above steps identify two stringsx1 and x2 which take equal values on the qubits indif qubits, and no other strings in S take the same ...

  6. [6]

    (a) If x1[dif ] ̸= 1, add a NOT gate to the line dif

    Circuit building: Initialize ˆU = ˆ1. (a) If x1[dif ] ̸= 1, add a NOT gate to the line dif . In this case, x1[dif ] is already 1. ˆU = dif → 0 1 2 3 (D1) (b) For all qubits b other than the dif qubit, if x1[b] ̸= x2[b], apply a CNOT gate on qubit b controlled on dif . ˆU = dif → 0 1 2 3 (D2) x1[dif ] = 1 and x2[dif ] = 0 by design. Thus, the application o...

  7. [7]

    ∞X n2=0 ⟨n2| ˆS(r) − ˆSΛ(r)|n1⟩ 2 #1/2 ≤ argmaxn1≤R

    T runcation error The goal is to compare the action of the full squeezing operator, ˆS(r), and the approximate squeezing operator, ˆSΛ(r) on the single- and multi-mode core states. The single-mode rank- R core state is given by |C⟩R = ¯P ncn |n⟩. Here, the sum ¯P is restricted to a sum of R 2 + 1 terms, such that |C⟩ is a symmetric rank- R core state as d...

  8. [8]

    T rotter error While we presented a state-dependent, i.e., ( R, Q)-dependent, bound for the truncation error, it is common and often more straightforward to establish a state-independent bound for the Trotter error. Thus, we will bound, |ψΛ⟩R − |ψΛ,K⟩R ≤ ˆSΛ,K(r) − ˆSΛ(r) , (E23) where ˆA denotes the spectral norm of the operator which is induced by the i...

Show all 151 references
  1. [9]

    (E33) Since the core state, as well as the squeezing operator(s), are characterized by real coefficients, the overlap ⟨ψ|ψΛ,K⟩ is real

    Combined errors The distance between the true state |ψ⟩R,Q and the state obtained by the application of the truncated, Trotterized squeezing operator(s) satisfies |ψ⟩R,Q − |ψΛ,K⟩R,Q 2 = 2 1 − Re(R,Q⟨ψ|ψΛ,K⟩R,Q) ≤ ϵ(N ) 2 . (E33) Since the core state, as well as the squeezing o...

  2. [10]

    (52) can be explicitly expressed as ˆsq,Λ m = ˆsq,Λ,+ m + ˆsq,Λ,− m , (F1) 44 where ˆsq,Λ,+ m := i 2 ⌊ Λ−2−m 4 ⌋X n=0 ℓ4n+m |qΛ(4n + m + 2)⟩ ⟨4n + m| , sq,Λ,− m := sq,Λ,+ m †

    Implementation of the squeezing operator using a singular-value decomposition Each operator ˆsq,Λ m with m ∈ {0, · · ·, 3} in Eq. (52) can be explicitly expressed as ˆsq,Λ m = ˆsq,Λ,+ m + ˆsq,Λ,− m , (F1) 44 where ˆsq,Λ,+ m := i 2 ⌊ Λ−2−m 4 ⌋X n=0 ℓ4n+m |qΛ(4n + m + 2)⟩ ⟨4n + ...

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    squeezing Hamiltonian

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