Pith. sign in

REVIEW 3 major objections 5 minor 64 references

Quantum phase estimation with near-optimal confidence intervals can run on a three-qubit control register.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-03 08:32 UTC pith:OJZIQQDK

load-bearing objection Solid numerical work on DPSS MPS preparation, but the three-qubit recycling protocol is asserted, not derived—the title claim hangs on it. the 3 major comments →

arxiv 2601.16474 v2 pith:OJZIQQDK submitted 2026-01-23 quant-ph

Quantum phase estimation with optimal confidence interval using three control qubits

Kaur Kristjuhan , Dominic W. Berry This is my paper
classification quant-ph
keywords quantum phase estimationdiscrete prolate spheroidal sequencematrix product stateconfidence intervalsemiclassical Fourier transformstate preparationT-gate costearly fault-tolerant quantum computing
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the discrete prolate spheroidal sequence (DPSS) state — the control-register state that yields the smallest confidence interval for a given confidence level in quantum phase estimation — can be approximated with very high accuracy by a matrix product state (MPS) with bond dimension 4. This matters because such an MPS is prepared by a sequence of operations on at most three qubits at a time, and when the dimension is a power of two the preparation can be interleaved with the semiclassical Fourier transform so the control register never holds more than three physical qubits. Numerical results for dimensions up to 2^24 and confidence levels through 99.99% show infidelity near 10^-8 for power-of-two dimensions and a relative increase in estimation error below 0.1% at practical confidence levels, so the near-optimal confidence-interval property of the DPSS is preserved. The paper also gives an explicit synthesis circuit whose T-gate count grows as O(n log(1/epsilon)), replacing ancilla-heavy arbitrary-state preparation with a small sequential circuit suited to early fault-tolerant hardware.

Core claim

The central claim is that the DPSS control state does not need to be prepared as a whole. Building it as a right-canonical MPS of bond dimension 4 decomposes the state into isometries that act on at most three qubits, applied from the most significant qubit downward. In the authors' numerical tests, the infidelity of this approximation sits below 10^-7 for six or more qubits at confidence levels 99–99.99% and stays essentially constant as the dimension is doubled all the way to 2^24; the relative increase in the error probability when the MPS replaces the exact DPSS is below 0.1% for confidence levels up to 99.99% and below one part in 10^11 at 99% for power-of-two dimensions. When the dimen

What carries the argument

The load-bearing object is an MPS of bond dimension 4 approximating the DPSS amplitude vector. In the right-canonical form used here, the state is produced by isometries M^(k), each acting on at most 1 + log2(chi) = 3 qubits and applied sequentially from the most significant qubit; this ordering is what makes the circuit compatible with the semiclassical Fourier transform, since a measured qubit can be reset and recycled for the next isometry. A second piece of machinery is the synthesis of these isometries: cosine-sine decomposition and a type-D orthogonal decomposition rewrite each M^(k) as single-qubit rotations and Clifford gates, yielding the explicit T-gate count. For dimensions that a

Load-bearing premise

The load-bearing premise is that interleaving the three-qubit MPS preparation with the measurements, resets, and classically controlled rotations of the semiclassical Fourier transform yields exactly the same phase-estimate distribution as preparing the full DPSS state and then applying the inverse QFT; the paper draws this from an analogy to the chi=2 case and gives no derivation for chi=4.

What would settle it

Calculate the chi=4 MPS approximation for a DPSS of dimension 2^25 and check whether the infidelity still lies on the ~10^-8 plateau seen up to 2^24; a jump would break the constant-bond-dimension claim. Alternatively, simulate (or run) the three-qubit recycling circuit of Figure 4 with an 8-qubit register and compare the empirical confidence-interval coverage with the ideal DPSS-plus-inverse-QFT prediction — a statistically significant mismatch would falsify the recycling equivalence.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For power-of-two dimensions, phase estimation with a confidence-interval-optimal control state can be run with just three physical qubits in the control register, making it feasible on early fault-tolerant hardware with very few logical qubits.
  • State-preparation T-gate count is O(n log(1/epsilon)) — approximately 9.52n log2(1/epsilon) + 82.62n - 24.64 log2(1/epsilon) - 213.84 — lower than prior DPSS preparation circuits and without the large ancilla overhead.
  • At confidence levels up to 99%, the relative increase in error probability from using the chi=4 MPS is at most one part in 10^11 for power-of-two dimensions, and it stays below 3x10^-4 at 99.99%, so the DPSS's confidence-interval guarantee is essentially preserved.
  • For non-power-of-two dimensions, the same MPS construction prepares the DPSS via padding and an inequality test, with a success probability close to 99% for the tested D=34 case; this reduces circuit complexity by up to half at the cost of qubit recycling.
  • The synthesis pipeline applies to any MPS with real-valued amplitudes, so the reduced gate count is available for other state-preparation tasks beyond phase estimation.

Where Pith is reading between the lines

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

  • If the periodic convergence observed through 2^24 is asymptotic, the chi=4 MPS accuracy should persist for arbitrarily large power-of-two dimensions, which would make the three-qubit protocol scalable to precision levels beyond those tested; a proof of this remains open.
  • The recycling argument is made for power-of-two dimensions only; for non-power-of-two dimensions the inequality test and unary iteration need access to the entire state, so extending the three-qubit scheme there would require a fundamentally different measurement schedule.
  • The same sequential MPS preparation could be adapted to other control states used in phase estimation — for example Kaiser-window or sine-window states — or to amplitude-estimation tasks, giving a general small-footprint estimation recipe with near-optimal confidence properties.
  • The remaining practical bottleneck for very large n is classical: computing the DPSS amplitudes and the MPS tensors at high precision. A dedicated classical algorithm for that step would let the method reach dimensions far beyond 2^24.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper addresses quantum phase estimation (QPE) with a control state chosen to optimize the confidence interval, namely the discrete prolate spheroidal sequence (DPSS) state. The authors propose preparing an approximation to this state using a matrix product state (MPS) with bond dimension 4, and report numerically that the infidelity is below roughly 9×10^-8 for all tested dimensions up to 2^24. For power-of-two dimensions, they claim that the MPS preparation circuit can be combined with the Griffiths–Niu semiclassical Fourier transform so that only three control qubits are ever active. For non-power-of-two dimensions, they use smooth padding from the continuous prolate spheroidal wave function followed by an inequality test. They also provide a circuit synthesis with O(n) single-qubit rotations, estimate a T-gate cost that scales as O(n log(1/ε)), compare it with prior state-preparation methods, and derive an analytic bound relating fidelity to the confidence-level degradation. The central resource claim is near-optimal-confidence QPE with a constant-size control register and O(n) T gates.

Significance. If the claims hold, the result is significant for early fault-tolerant quantum computation: it removes the exponential state-preparation overhead usually associated with DPSS states and reduces the control-register width to a small constant, which is directly relevant to algorithms with limited logical qubits. The paper is commendably explicit about the numerical nature of the central evidence, and the analytic bound in Appendix A plus the detailed gate decompositions strengthen the presentation. However, the three-control-qubit claim rests on an equivalence between the interleaved MPS-recycling circuit and the standard full-state QPE circuit, and that equivalence is asserted rather than derived. The paper also ships no code or data, which makes the non-power-of-two interpolation results hard to verify independently. These are the main gaps between the current manuscript and a fully supported claim.

major comments (3)
  1. [Sec. II D, Fig. 4] The three-qubit recycling protocol is the basis of the title, but it is justified only by an analogy to Ref. [13] and by citing Ref. [41]. The Griffiths–Niu construction replaces the inverse QFT of a fully prepared register with measurements and classically controlled rotations; it does not by itself justify interleaving the MPS isometries M[k] with those measurements. After the first measurement, the remaining bond degrees of freedom are in a conditional state, and one must show that resetting and reusing the measured qubit for later M[k] reproduces the same joint statistics as preparing the full DPSS state and then applying the inverse QFT. Please provide a self-contained argument, or a precise statement and proof sketch of the circuit identity, for the χ=4 case. Without this, the 'three control qubits' claim is unsupported.
  2. [Sec. II E and Fig. 7] For non-power-of-two dimensions, the numerical results for n≥17 depend on interpolated SpheroidalPS padding amplitudes. The text argues that the padding values are unimportant because they are discarded by the inequality test, but the padding amplitudes influence the MPS truncation error inside the support, as the paper's own comparison with zero-amplitude padding shows. No interpolation error analysis or data are given. Please report convergence checks against direct evaluations at representative high dimensions, or release the padding data and scripts, so that the general-dimension claim is reproducible.
  3. [Sec. III and Sec. V] The claim that bond dimension 4 remains sufficient 'for any dimension' is an extrapolation from n≤24. The authors are honest about this in the conclusion, but the same statement is presented more assertively earlier in the paper and in the abstract's framing. Since the extrapolation is load-bearing for the practical scope of the method, the paper should either state the dimension-independence as an explicit conjecture supported by the convergence data, or provide a more quantitative argument that the observed periodic pattern provably persists. As written, the 'any dimension' formulation goes beyond what the numerical evidence alone establishes.
minor comments (5)
  1. [Sec. II A, Eq. (2)] The normalization of Γ uses 1/√N even though the sum runs over D<N terms. This is correct because the state is implicitly zero-padded to N, but the zero-padding should be stated explicitly at that point.
  2. [Sec. II D] The phrase 'most significant qubit measured first, yielding the least significant bit of the phase estimate' is potentially confusing because bit-ordering conventions vary. Please specify the convention used (e.g., whether the phase is read as a binary fraction from the first or last measured bit).
  3. [Table I] For n=5, the infidelity is reported as exactly 0. It would help to state explicitly that this is exact because a bond dimension of 4 can represent any 5-qubit state, rather than a numerical artifact.
  4. [Sec. IV, Eq. (29)] The T-gate formula is useful, but the reader has to reverse-engineer how the special cases fM[2] and the first operator are included. A one-sentence explanation that the formula already accounts for those reductions would improve clarity.
  5. [General] No code, data, or data-availability statement is provided. Given that the central results are numerical, even a small repository with the MPS construction and the padding interpolation would substantially aid reproducibility.

Circularity Check

0 steps flagged

No circular reduction found; main numerical claims are tested against an external DPSS target, though the three-qubit recycling protocol is supported by analogy to the authors' prior χ=2 work and lacks a derivation.

full rationale

The paper's central numerical claim—that a bond-dimension-4 MPS gives a high-fidelity approximation to the DPSS state—is checked against externally defined DPSS amplitudes obtained from SciPy's implementation of the Slepian tridiagonal eigenvector formulation. The fidelity and confidence-level evaluations use the standard S(d) functional from prior literature (Eq. 22) and are not used to define or fit the target state. The Appendix A bound is derived from fidelity and the Bhattacharyya inequality rather than assumed as an input. The only flagged weakness is Section II D / Figure 4: the three-qubit recycling protocol is asserted by stating that the χ=4 circuit is 'analogous to the circuits of Ref. [13]' and that 'we can delay the application of each M[k] until a previously used qubit is made available after performing a measurement.' No equation shows that the interleaved MPS isometries commute with the semiclassical measurements for χ=4. This is a genuine correctness gap and a load-bearing unsupported step, but it is not a circular reduction: the paper does not fit a parameter to a target and then rename it a prediction, nor does it define its conclusion in terms of its input. The conclusion's explicit admission that the arbitrary-dimension accuracy claim lacks a formal proof is an honest limitation, not circularity. I therefore assign a low score reflecting the one self-citation/analogy gap while recognizing that the main numerical derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central claims rest on prior DPSS optimality, an unproven interleaving of semiclassical QFT with MPS preparation, numerical extrapolation beyond n=24, and numerically synthesized rotation angles. No new physical entities are introduced.

free parameters (3)
  • MPS bond dimension chi = 4
    Selected by inspecting fidelity vs. cost in Table I; no derivation proves chi=4 suffices for all dimensions.
  • Padding amplitudes for non-power-of-two dimensions = Samples/interpolation of prolate spheroidal wave function P0_0(x/N, dN/2)
    Chosen because zero padding gives much worse MPS approximation; interpolation used for large D. This is an ad hoc modeling choice.
  • Circuit rotation angles for each fM[k] = Obtained via scipy.optimize.minimize; Frobenius distance < 1e-10
    Angles are fit numerically for each isometry; no analytic formula is provided for general n.
axioms (6)
  • domain assumption DPSS state is the optimal control state for confidence-interval QPE (Imai-Hayashi [15], Slepian [16]).
    Basis for the entire protocol; standard result imported from prior literature.
  • domain assumption Griffiths-Niu semiclassical QFT can be interleaved with MPS preparation and qubit recycling for arbitrary control amplitudes.
    Section II D and Figure 4 assume the measurement-and-classical-rotation pattern is equivalent to inverse QFT; cited [41] and analogous to [13], but not proven here.
  • ad hoc to paper Numerical results for n <= 24 extrapolate to all practical dimensions.
    Section III C and Conclusion: no formal proof; authors state the open question. Convergence evidence (Figure 8) is heuristic.
  • standard math 2-by-1 CSD and type-D Cartan decomposition cover all real isometries used.
    Section IV relies on these standard matrix decompositions.
  • ad hoc to paper Numerical optimization can synthesize the rotation angles for every fM[k].
    Section IV: tested on DPSS and random orthogonal matrices with Frobenius distance < 1e-10, but no constructive analytic synthesis or shipped code.
  • ad hoc to paper For non-power-of-two D, padding with the prolate spheroidal wave function yields an MPS that, after the inequality test, retrieves the DPSS state with high success probability.
    Section II E; the padding choice is justified empirically, not derived.

pith-pipeline@v1.3.0-alltime-deepseek · 20274 in / 14142 out tokens · 154570 ms · 2026-08-03T08:32:48.020550+00:00 · methodology

0 comments
read the original abstract

Quantum phase estimation is an important routine in many quantum algorithms, particularly for estimating the ground state energy in quantum chemistry simulations. This estimation involves applying powers of a unitary to the ground state, controlled by an auxiliary state prepared on a control register. In many applications the goal is to provide a confidence interval for the phase estimate, and optimal performance is provided by a discrete prolate spheroidal sequence. We show how to prepare the corresponding state in a far more efficient way than prior work. We find that a matrix product state representation with a bond dimension of 4 is sufficient to give a highly accurate approximation for all dimensions tested, up to $2^{24}$. This matrix product state can be efficiently prepared using a sequence of simple three-qubit operations. When the dimension is a power of 2, the phase estimation can be performed with only three qubits for the control register, making it suitable for early-generation fault-tolerant quantum computers with a limited number of logical qubits.

Figures

Figures reproduced from arXiv: 2601.16474 by Dominic W. Berry, Kaur Kristjuhan.

Figure 1
Figure 1. Figure 1: FIG. 1. Variant of the first steps of the QPE algorithm with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diagram (a) illustrates an operator [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Quantum circuit for preparing an MPS with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Circuit for performing QPE for a unitary operator [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Circuit for preparing a [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Plot (a) shows the state [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Relative increase in error probability [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Infidelity between [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Confidence interval half-width [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Circuits obtained by applying a series of useful decompositions to facilitate synthesis of an [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Two possible circuits for synthesising one operator [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗

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

64 extracted references · 10 linked inside Pith

  1. [1]

    P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing41, 303 (1999)

    1999

  2. [2]

    As the dimension is increased, the performance improves until it is best again at the next power of 2

    The performance is best for dimensions that are a power of 2, and worst if the dimension is one more than a power of 2. As the dimension is increased, the performance improves until it is best again at the next power of 2. We expect that this behaviour is because doubling the dimension corresponds to approximating the same function, but with one extra les...

  3. [3]

    Circuit (c) is obtained by mergingM [k−1] with M [k] a in circuit (b)

    Circuit (b) is obtained by applying the CSD to (a). Circuit (c) is obtained by mergingM [k−1] with M [k] a in circuit (b). Circuit (d) is obtained by performing a type D Cartan decomposition toM [k] b in circuit (c). This sequence of decompositions (a)-(d) can be sequentially applied for each value ofkto obtain a decomposed circuit for the entire MPS. We ...

  4. [4]

    Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferov´ a, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. Sawaya,et al., Quantum chemistry in the age of quantum computing, Chemical Reviews119, 10856 (2019)

    2019

  5. [5]

    Aspuru-Guzik, A

    A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Simulated quantum computation of molecular energies, Science309, 1704 (2005)

    2005

  6. [6]

    McArdle, S

    S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Quantum computational chemistry, Reviews of Modern Physics92, 015003 (2020)

    2020

  7. [7]

    A. M. Dalzell, S. McArdle, M. Berta, P. Bienias, C.-F. Chen, A. Gily´ en, C. T. Hann, M. J. Kastoryano, E. T. Khabiboulline, A. Kubica,et al., Quantum algorithms: A survey of applications and end-to-end complexities, arXiv:2310.03011 (2023)

    Pith/arXiv arXiv 2023

  8. [8]

    Bauer, S

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

    2020

  9. [9]

    Motta and J

    M. Motta and J. E. Rice, Emerging quantum computing algorithms for quantum chemistry, Wiley Interdisciplinary Reviews: Computational Molecular Science12, e1580 (2022)

    2022

  10. [10]

    Poulin, A

    D. Poulin, A. Kitaev, D. S. Steiger, M. B. Hastings, and M. Troyer, Quantum algorithm for spectral measurement with a lower gate count, Phys. Rev. Lett.121, 010501 (2018)

    2018

  11. [11]

    Alternatively, the angles could be directly calculated by explicitly performing the decompositions and subsequent ZX-calculus based optimisation procedure on fM [k]

    We find that for both the MPS matrices for a DPSS state, as well as for randomly generated orthogonal matrices, the optimisation procedure is successful, achieving a Frobenius distance below 10−10 regardless of how the initial values of the angles are chosen. Alternatively, the angles could be directly calculated by explicitly performing the decomposition...

    2024

  12. [12]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(Cambridge uni- versity press, 2010)

    2010

  13. [13]

    D. W. Berry, M. Kieferov´ a, A. Scherer, Y. R. Sanders, G. H. Low, N. Wiebe, C. Gidney, and R. Bab- bush, Improved techniques for preparing eigenstates of fermionic hamiltonians, npj Quantum Infor- mation4, 22 (2018)

    2018

  14. [14]

    Luis and J

    A. Luis and J. Peˇ rina, Optimum phase-shift estimation and the quantum description of the phase difference, Physical Review A54, 4564 (1996)

    1996

  15. [15]

    Babbush, C

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear T complexity, Physical Review X8, 041015 (2018)

    2018

  16. [16]

    Najafi, P

    P. Najafi, P. Costa, and D. W. Berry, Optimum phase estimation with two control qubits, A VS Quantum Science5(2023)

    2023

  17. [17]

    D. W. Berry, Y. Tong, T. Khattar, A. White, T. I. Kim, G. H. Low, S. Boixo, Z. Ding, L. Lin, S. Lee, et al., Rapid initial-state preparation for the quantum simulation of strongly correlated molecules, PRX Quantum6, 020327 (2025)

    2025

  18. [18]

    Imai and M

    H. Imai and M. Hayashi, Fourier analytic approach to phase estimation in quantum systems, New Journal of Physics11, 043034 (2009)

    2009

  19. [19]

    Slepian, Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case, Bell System Technical Journal57, 1371 (1978)

    D. Slepian, Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case, Bell System Technical Journal57, 1371 (1978)

    1978

  20. [20]

    Kaiser and R

    J. Kaiser and R. Schafer, On the use of the I 0-sinh window for spectrum analysis, IEEE Transactions on Acoustics, Speech, and Signal Processing28, 105 (1980). 22

    1980

  21. [21]

    D. W. Berry, Y. Su, C. Gyurik, R. King, J. Basso, A. D. T. Barba, A. Rajput, N. Wiebe, V. Dun- jko, and R. Babbush, Analyzing prospects for quantum advantage in topological data analysis, PRX Quantum5, 010319 (2024)

    2024

  22. [22]

    Grover and T

    L. Grover and T. Rudolph, Creating superpositions that correspond to efficiently integrable probability distributions, arXiv:quant-ph/0208112 (2002)

    Pith/arXiv arXiv 2002

  23. [23]

    G. H. Low, V. Kliuchnikov, and L. Schaeffer, Trading T gates for dirty qubits in state preparation and unitary synthesis, Quantum8, 1375 (2024)

    2024

  24. [24]

    Greenaway, W

    S. Greenaway, W. Pol, and S. Sim, A case study against QSVT: assessment of quantum phase esti- mation improved by signal processing techniques, arXiv:2404.01396 (2024)

    Pith/arXiv arXiv 2024

  25. [25]

    O’Brien and C

    O. O’Brien and C. S¨ underhauf, Quantum state preparation via piecewise QSVT, Quantum9, 1786 (2025)

    2025

  26. [26]

    McArdle, A

    S. McArdle, A. Gily´ en, and M. Berta, Quantum state preparation without coherent arithmetic, arXiv:2210.14892 (2022)

    Pith/arXiv arXiv 2022

  27. [27]

    Patel, S

    D. Patel, S. J. S. Tan, Y. Subasi, and A. T. Sornborger, Optimal coherent quantum phase estimation via tapering, arXiv:2403.18927 (2024)

    Pith/arXiv arXiv 2024

  28. [28]

    G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Physical Review Letters118, 010501 (2017)

    2017

  29. [29]

    J. Lee, D. W. Berry, C. Gidney, W. J. Huggins, J. R. McClean, N. Wiebe, and R. Babbush, Even more efficient quantum computations of chemistry through tensor hypercontraction, PRX Quantum 2, 030305 (2021)

    2021

  30. [30]

    D. W. Berry, M. J. Hall, M. Zwierz, and H. M. Wiseman, Optimal heisenberg-style bounds for the average performance of arbitrary phase estimates, Physical Review A—Atomic, Molecular, and Optical Physics86, 053813 (2012)

    2012

  31. [31]

    F. J. Simons, Slepian functions and their use in signal estimation and spectral analysis, inHandbook of geomathematics(Springer, 2010) pp. 891–923

    2010

  32. [32]

    Vidal, Efficient classical simulation of slightly entangled quantum computations, Physical Review Letters91, 147902 (2003)

    G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Physical Review Letters91, 147902 (2003)

    2003

  33. [33]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics326, 96 (2011)

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics326, 96 (2011)

    2011

  34. [34]

    C. M. Dawson and M. A. Nielsen, The Solovay-Kitaev algorithm, Quantum Info. Comput.6, 81–95 (2006)

    2006

  35. [35]

    T. Hao, A. Xu, and S. Tannu, Reducing T gates with unitary synthesis, arXiv:2503.15843 (2025)

    arXiv 2025

  36. [36]

    M. Amy, D. Maslov, M. Mosca, and M. Roetteler, A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems32, 818 (2013)

    2013

  37. [37]

    Gheorghiu, M

    V. Gheorghiu, M. Mosca, and P. Mukhopadhyay, T-count and T-depth of any multi-qubit unitary, npj Quantum Information8, 141 (2022)

    2022

  38. [38]

    Paradis, J

    A. Paradis, J. Dekoninck, B. Bichsel, and M. Vechev, Synthetiq: Fast and versatile quantum circuit synthesis, Proceedings of the ACM on Programming Languages8, 55 (2024)

    2024

  39. [39]

    Weiden, J

    M. Weiden, J. Kalloor, E. Younis, J. Kubiatowicz, and C. Iancu, High precision fault-tolerant quantum circuit synthesis by diagonalization using reinforcement learning, arXiv:2409.00433 (2024)

    Pith/arXiv arXiv 2024

  40. [40]

    R. Iten, R. Colbeck, I. Kukuljan, J. Home, and M. Christandl, Quantum circuits for isometries, Physical Review A93, 032318 (2016)

    2016

  41. [41]

    Szasz, E

    A. Szasz, E. Younis, and W. De Jong, Numerical Circuit Synthesis and Compilation for Multi-State Preparation, in2023 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Bellevue, W A, USA, 2023) pp. 768–778

    2023

  42. [42]

    Fomichev, K

    S. Fomichev, K. Hejazi, M. S. Zini, M. Kiser, J. Fraxanet, P. A. M. Casares, A. Delgado, J. Huh, A.-C. Voigt, J. E. Mueller,et al., Initial state preparation for quantum chemistry on quantum computers, PRX Quantum5, 040339 (2024)

    2024

  43. [43]

    W. J. Huggins, O. Leimkuhler, T. F. Stetina, and K. B. Whaley, Efficient state preparation for the quantum simulation of molecules in first quantization, PRX Quantum6, 020319 (2025)

    2025

  44. [44]

    R. B. Griffiths and C.-S. Niu, Semiclassical Fourier transform for quantum computation, Physical Review Letters76, 3228 (1996)

    1996

  45. [45]

    com/language/ref/SpheroidalPS.html

    Wolfram Research (2007), SpheroidalPS, Wolfram Language function,https://reference.wolfram. com/language/ref/SpheroidalPS.html. 23

    2007

  46. [46]

    DLMF,NIST Digital Library of Mathematical Functions,https://dlmf.nist.gov/, Release 1.2.3 of 2024-12-15, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

    2024

  47. [47]

    Wolfram Research, Inc., Mathematica, Version 14.3, Champaign, IL, 2025

    2025

  48. [48]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. May- orov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henrikse...

    2020

  49. [49]

    R. M. Grayet al., Toeplitz and circulant matrices: A review, Foundations and Trends®in Commu- nications and Information Theory2, 155 (2006)

    2006

  50. [50]

    Oppenheim, R

    A. Oppenheim, R. Schafer, and J. Buck,Discrete-time signal processing, Prentice Hall International Editions Series (Prentice Hall, 1999)

    1999

  51. [51]

    G. H. Golub and C. F. Van Loan,Matrix computations(Math. Sci., Johns Hopkins University Press, Baltimore, MD, 1996)

    1996

  52. [52]

    C. C. Paige and M. Wei, History and generality of the CS decomposition, Linear Algebra and its Applications208, 303 (1994)

    1994

  53. [53]

    B. D. Sutton, Computing the complete CS decomposition, Numerical Algorithms50, 33 (2009)

    2009

  54. [54]

    Wierichs, M

    D. Wierichs, M. West, R. T. Forestano, M. Cerezo, and N. Killoran, Recursive Cartan decompositions for unitary synthesis, arXiv:2503.19014 (2025)

    Pith/arXiv arXiv 2025

  55. [55]

    Wei and Y.-M

    H.-R. Wei and Y.-M. Di, Decomposition of orthogonal matrix and synthesis of two-qubit and three- qubit orthogonal gates, Quantum Info. Comput.12, 262–270 (2012)

    2012

  56. [56]

    V. V. Shende, S. S. Bullock, and I. L. Markov, Synthesis of Quantum Logic Circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems25, 1000 (2006), quant-ph/0406176

    Pith/arXiv arXiv 2006

  57. [57]

    Kissinger and J

    A. Kissinger and J. van de Wetering, PyZX: Large scale automated diagrammatic reasoning, arXiv:1904.04735 (2019)

    Pith/arXiv arXiv 1904

  58. [58]

    Duncan, A

    R. Duncan, A. Kissinger, S. Perdrix, and J. Van De Wetering, Graph-theoretic simplification of quan- tum circuits with the ZX-calculus, Quantum4, 279 (2020). [56]https://github.com/zxcalc/pyzx

    2020

  59. [59]

    Perdomo, N

    O. Perdomo, N. Castaneda, and R. Vogeler, Preparation of 3-qubit states, arXiv:2201.03724 (2022)

    Pith/arXiv arXiv 2022

  60. [60]

    Kliuchnikov, K

    V. Kliuchnikov, K. Lauter, R. Minko, A. Paetznick, and C. Petit, Shorter quantum circuits via single- qubit gate approximation, Quantum7, 1208 (2023)

    2023

  61. [61]

    D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization, Quantum3, 208 (2019)

    2019

  62. [62]

    Gidney, Halving the cost of quantum addition, Quantum2, 74 (2018)

    C. Gidney, Halving the cost of quantum addition, Quantum2, 74 (2018)

    2018

  63. [63]

    Gidney and A

    C. Gidney and A. G. Fowler, Efficient magic state factories with a catalyzed|CCZ⟩to 2|T⟩transfor- mation, Quantum3, 135 (2019)

    2019

  64. [64]

    Watrous,The theory of quantum information(Cambridge university press, 2018)

    J. Watrous,The theory of quantum information(Cambridge university press, 2018). Appendix A: Derivation of bounds SinceS(d)≡ d π P j,k sinc(d(j−k))|j⟩ ⟨k|is a positive semi-definite operator, then{S(d),11−S(d)} is a positive operator valued measure and we can use a well-known inequality relating the fidelity of quantum states to the Bhattacharyya coefficie...

    2018