Mid-circuit measurements and feedforward implement every unitary exactly in constant depth

Chenfeng Cao

REVIEW

Mid-circuit measurements and feedforward implement every unitary exactly in constant depth

Chenfeng Cao

Quantum Physics · Submitted 5 Oct 2026 · 50 references

Pith's verdict: The residual-repair identity is correct and new, but Theorem 1's sharp width and constant depth both rest on two unverified 2026 preprints; exactness survives independently only at exponentially larger width.

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arXiv ID2610.06551 (version 1)

Pith IDLNQO3TUV, Pith's permanent record of this paper and its review

CitationChenfeng Cao. "Mid-circuit measurements and feedforward implement every unitary exactly in constant depth." arXiv:2610.06551 (2026). https://pith.science/paper/LNQO3TUV

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Editor's note

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Two things to know. First, the core idea is genuinely good and the central algebra checks out: I redid the Theorem 2 expansion (Eq. 9) and it is correct, and the condition nu <= 1 follows from the operator-norm hypothesis rather than being fitted. The residual-repair trick — approximate at error 1/d, teleport the residual, add it back coherently, one amplification step — is new against the cited prior art and is likely durable. Second, the headline resource claim is conditional, and the paper says so itself. Theorem 2 states the residual-preparation as a hypothesis rather than proving it, and Lemma 3's circuit implementation, including the exact constant-depth Fourier circuit (Lemma S3, explicitly attributed to Nehoran–Yuen's Algorithm 6), is borrowed from an unpublished 2026 preprint and not reproduced here. Lemma 4's state preparation similarly leans on Strelchuk et al. If those two preprints fail, the width O(n d^2 log(n+2)) collapses, and the paper's own independent route (Supplementary G) is at width 2^{O(2^n)} — exponentially larger. So the exact-constant-depth existence statement survives independently; the sharp width does not yet.

What the paper does well beyond the core identity: the lower bound (Prop. 5) is standard parameter counting and fine; the fanout-by-feedforward gadget in Supp. D has a careful branch-by-branch analysis that reads correctly; the writeup is honest about the two unresolved width questions and about the AI contribution; and Supplementary G is a real independent fallback, not a hand-wave. The AI disclosure deserves one parenthetical: the author claims independent verification plus small numerical checks, which is consistent with the algebra checking out, but it raises the bar for checking the parts that are cited rather than proved.

Where I land: this deserves a serious referee — yes, send it out. It answers Rosenthal's Question 1.2 and parts of Open Problem 17 conditional on [20] and [21], and the technique is valuable whether or not those preprints hold. A referee should not certify the width claim until both preprints are verified; a clean option is to accept with the resource statement hedged to Supplementary G's width, or to demand verification before publication. Anyone working on depth complexity or QAC0 should read it, and the residual-repair identity is quotable on its own. I'd cite it.

Referee report

Significance

If Lemma 3 and Lemma 4 hold at their stated widths, Theorem 1 is a major result: exact worst-case unitary synthesis in constant quantum depth, with width matching the parameter-counting lower bound of Proposition 5 up to polynomial-in-n factors, and it would settle the exact versions of several open questions cited in the Introduction. The strengths that I can confirm directly: the residual-repair identity (Eq. (9)) is correct — the expansion uses ΠVΠ = (U/2)⊗|0⟩⟨0| and ΠV†ΠVΠ = Π/4, verified algebraically — and the smallness condition ν ≤ 1 is derived from ‖U − Ũ‖_op ≤ 1/d rather than tuned, so the construction has no fitted parameter. The discretization error analysis of Supplementary A (Lemmas S1–S2) is self-contained, and the Gaussian-integral reduction (S2) checks out. The lower-bound arguments are standard and correctly assembled. What is not yet established by the manuscript itself is the circuit-level width claim: Lemma S3 and the leading framework of Lemma 4 are quoted from the unpublished preprints [21] and [20].

Major objections (2)

[§4.1, Lemma 3; Supplementary A, Lemma S3] Lemma 3, the constant-depth approximate circuit behind Theorem 1, and its width are imported from an unpublished preprint, so the headline width of Theorem 1 is conditional. Theorem 1's width O(nd² log(n+2)) is obtained in §4.1 from Lemma 3 at ε = 1/d (width O(d² log²(n+2))), combined with the residual-state width of Lemma 4 ([b286] states this width combination). The Gaussian identity and the discretization error bound (Lemmas S1–S2, Eq. (S9)) are self-contained in Supplementary A and, as far as I checked, correct. The circuit implementation, however, is not: Lemma S3 states that the small Fourier transform F_K has an exact clean constant-depth implementation on O(K⁴) qubits and is attributed to "Nehoran–Yuen's exact Fourier circuit [21, Sec. 5.3.1, Algorithm 6]", and the O(mK⁴) Fourier-layer width in the proof of Lemma 3 inherits that bound. Reference [21] (arXiv:2609.40351v1) is an unreviewed 2026 preprint whose Algorithm 6 is not reproduced here, so neither the depth nor the width of F_K can be checked from the manuscript. The authors acknowledge the dependence: the Discussion states that the independent route of Supplementary G has width 2^{O(2^n)} ([b439]), exponentially larger than the claimed O(n4^n log(n+2)). This is load-bearing: if [21]'s Fourier circuit has non-constant depth or width ω(K⁴), Lemma 3 breaks and Theorem 1's width collapses. Please reproduce a self-contained exact constant-depth F_K construction with the stated O(K⁴) width (plausible from the published ingredients [28, Lemma 4.13] and [47, Sec. 4.1]), or state Lemma 3 and Theorem 1 as conditional on [21].
[Lemma 4; Supplementary B; Theorem 2] The exact state preparation that supplies the residual-state width in Theorem 2 rests on the unpublished preprint [20]. Theorem 2's hypothesis ([b79]) that the 2n-qubit state proportional to Σ_{i,j} E_{ij}|i⟩|j⟩ has an exact clean preparation at width w_E is used with q = 2n in §4.1: Lemma 4 gives w_E = O(q2^q) = O(nd²), which is the leading width term of Theorem 1. The proof of Lemma 4 in Supplementary B is framed as a reduction of "the O(4^q) workspace of Ref. [20, Theorem 4.2]" to O(q2^q) by sharing prefix computations; the first-one construction is that of Ref. [20, Theorem 4.2] (arXiv:2609.40294v1, unpublished). The key resource assertion — that all N prefix predicates g_k "can all be computed cleanly in constant depth on O(qN) qubits with generalized Toffoli gates" — is supported only by a sketch (block decomposition (S17) and two copying stages) without a stage-by-stage depth and width ledger. This matters because Theorem 1's depth is O(1): if any copying or clearing stage has logarithmic depth, or if [20]'s base width is larger than cited, both the depth and the width of Theorem 1 change. Please make Lemma 4 fully self-contained, with explicit constants and an explicit per-stage width and depth accounting.

Minor objections (6)

[Abstract and Introduction] Several formulas lost their superscript formatting in the arXiv text (for example "Ford= 2 n", "~O(d2 log4(d/ε))", and "Ω(4 n/(h+1))"); a careful typesetting pass is needed before publication.
[§4.1] The sentence "Take the approximate circuit of Lemma 3 with ε = 1/d; its width is O(d² log²(n+2))" silently drops the second term d log⁴(d/ε) of Lemma 3, which at ε = 1/d equals 16 d n⁴; the drop is valid only asymptotically since d = 2^n, and stating the dominance condition explicitly would help the reader.
[§4.2, Eq. (10)] The width bound w_meas ≤ 3w is asserted without derivation; a one-line justification from the gadget of Supplementary D (2q+1 ancillas per q-target fanout, reused after reset) would make the factor 3 transparent.
[Lemma 3 and Supplementary A] The notation "log⁴(d/ε)" is easy to misread as an iterated logarithm, especially since the manuscript also writes "log log(d/ε)"; a sentence defining log⁴ x = (log x)⁴ at first use would prevent confusion.
[Fig. 1] The caption references registers A, B, prep, selector, and flag, and operators R0, PE, C, and CE, but not all of these are marked in the drawing; the caption should identify every symbol it uses.
[Supplementary G and Discussion] The claim that Supplementary G is "independent of Nehoran–Yuen" is accurate only up to Lemma 4 and Lemma S4, which still use the framework of Refs. [20] and [47]; the Discussion should state which dependencies remain after the independence claim.

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The paper

The paper claims

Mid-circuit measurements and feedforward reduce exact unitary synthesis depth from Θ(n) to O(1), at the price of O(n 4^n log n) ancillas.

The reading

An arbitrary n-qubit quantum operation can, in principle, be carried out exactly in a constant number of parallel layers of one- and two-qubit gates, provided the machine may measure qubits mid-circuit and steer later gates with the classical outcomes. The paper proves that measuring with feedforward drops the optimal worst-case depth for exact unitary synthesis from Θ(n) to O(1), using O(n 4^n log(n+2)) total qubits, every ancilla starting and ending in |0⟩. Depth, not gate count, is the scarce resource on parallel quantum hardware, so a constant-depth exact implementation reshapes what a wide, shallow device can do. The same width yields an exact clean constant-depth fanout circuit, and generic unitaries provably need Ω(4^n) qubits at constant depth, so the construction is near-optimal in width.

What carries the argument

The carrying object is residual repair (Theorems 1–2): the residual E = U − Ũ of a constant-depth approximate circuit Ũ, encoded exactly as the 2n-qubit state proportional to Σ_{i,j} E_{ij} |i⟩|j⟩ and teleported onto the data to produce the block E/ν with ν = √d ∥E∥_F ≤ 1. A selector qubit coherently superposes the approximate block Ũ/(1+ν) with the teleported residual E/(1+ν), so the two interfere to exactly U/(1+ν); a flag rotates that to U/2, and one step of exact amplitude amplification, using the reflection identity V_ex = −V(I−2Π)V†(I−2Π)V, turns the success angle from π/6 to π/2, making success certain. The approximate circuit Ũ comes from the cited Gaussian, five-layer constant-depth construction at width O(d² log²(n+2)), and the exact preparation of the residual state uses a prefix-computation construction of width O(q 2^q).

Extended reading notes

Core claim

The paper's central claim is that every n-qubit unitary admits an exact, deterministic, constant-depth implementation built from one- and two-qubit gates, single-qubit measurements, O(1) rounds of parity feedforward, and O(n 4^n log(n+2)) qubits, with all ancillas starting and ending in |0⟩. The mechanism is 'residual repair': instead of teleporting the target U through its Choi state, which would force Θ(d) rounds of amplitude amplification because the identity-outcome block is U/d, the construction teleports only the matrix error E = U − Ũ of a constant-depth approximate circuit Ũ. Because ∥U − Ũ∥_op ≤ 1/d, the normalized residual E/ν with ν = √d ∥E∥_F ≤ 1 has exactly the size one amplification step can fix, so coherently combining Ũ with the teleported E/ν yields a block proportional to U, a flag rescales it to U/2, and a single exact amplification step makes success certain on every input superposition. This settles, for exact synthesis, the ordinary and fanout versions of the open problems posed in the cited literature, and the parameter-counting lower bound shows that Haar-almost every unitary still requires Ω(4^n) qubits at constant depth even with an unrestricted classical controller.

Load-bearing premise

The exactness and claimed width of the main theorem rest on the cited approximate synthesis of Ref. [21], which bounds the width of a constant-depth fanout circuit whose block approximates U to operator norm 1/d using O(d² log²(n+2)) qubits, and which the paper cites rather than reproduces; Theorem 2 additionally assumes the 2n-qubit residual state has an exact clean constant-depth preparation at stated width, a fact attributed to Refs. [18] and [20]. If either width bound fails, the width claim of Theorem 1 breaks, even though Supplementary Material G offers an independent exact route at doubly exponential width.

Editorial extensions

If this is right

Exact constant-depth synthesis makes feedforward, not measurement itself, the enabling resource: measurements without feedforward remain subject to the circuit causal-cone bound, so generic targets still require depth Ω(n) even with resets.
A chain of N qubits with a depth-O(log N) nearest-neighbour circuit can be rewritten as an exact constant-depth all-to-all circuit with parity feedforward and poly(N) qubits, covering O(log N) periods of a one-dimensional Floquet circuit.
Depth–width trade-offs follow from the construction: for 1 ≤ h ≤ 2^n, every unitary has an exact clean fanout or adaptive circuit of depth O(h) and width O((d²/h) log²(4d/h)), within O(log²(4d/h)) of the counting lower bound.
Generic targets require Ω(4^n/(h+1)) qubits at depth h even under unrestricted classical control, so the constant-depth construction meets the fundamental parameter-counting bound within polynomial factors.
A constant-accuracy coherent approximation suffices for exact synthesis: O(n) error-1/3 approximate circuits can be combined in parallel and the final residual corrected, and an independent proof at doubly exponential width avoids the approximate theorem entirely.

Reading between the lines

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

Residual repair is a transferable principle: any constant-depth approximate synthesis family with error at most 1/d can be upgraded to an exact synthesis at the cost of preparing a single residual state, so improvements to the approximation's width or error bound directly improve the exact construction.
The width claim of the main theorem inherits the assumptions of the cited approximate synthesis and of the exact residual-state preparation; a fully self-contained proof at the claimed width is the natural next target, since the independence route in Supplementary Material G comes at doubly exponential width.
Because practical constant-depth utility is limited by the O(n 4^n log n) width, the result matters most as an asymptotic landmark and as a template, and the open question of whether width can be reduced to ~O(d²) at constant depth or with a single feedforward round determines whether the construction has concrete hardware reach.

Original abstract

Mid-circuit measurements with feedforward reduce the optimal worst-case quantum depth for exact $n$-qubit unitary synthesis from $\Theta(n)$ to $O(1)$, using arbitrary one- and two-qubit gates and unrestricted ancillary space. Measurements and resets without feedforward remain subject to the circuit causal-cone bound. Our deterministic construction uses single-qubit measurements, $O(1)$ rounds of parity feedforward, and $O(n4^n\log(n+2))$ total qubits, with all ancillas starting and ending in $|0\rangle$. Classical processing is not counted in quantum depth. We teleport only the matrix error of an approximation by Nehoran and Yuen. An exactly prepared ancillary state encodes this error, whose action is added back by coherent interference. At operator-norm error $2^{-n}$, one step of amplitude amplification suffices. Coherent fanout circuits achieve the same constant-depth bound, while circuits using only one- and two-qubit gates attain depth $\Theta(n)$ with $O(4^n\log^2(n+2))$ gates. Generic targets require $\Omega(4^n/(h+1))$ qubits at depth $h$, even under unrestricted classical control. Our depth-width trade-offs meet this bound within a factor $O(n^2)$ for $1\le h\le2^n$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-06: 1 request, 7,586 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.44 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Mid-circuit measurements with feedforward reduce the optimal worst-case quantum depth for exact $n$-qubit unitary synthesis from $\Theta(n)$ to $O(1)$, using arbitrary one- and two-qubit gates and unrestricted ancillary space. Measurements and resets without feedforward remain subject to the circuit causal-cone bound. Our deterministic construction uses single-qubit measurements, $O(1)$ rounds of parity feedforward, and $O(n4^n\log(n+2))$ total qubits, with all ancillas starting and ending in $|0\rangle$. Classical processing is not counted in quantum depth. We teleport only the matrix error of an approximation by Nehoran and Yuen. An exactly prepared ancillary state encodes this error, whose action is added back by coherent interference. At operator-norm error $2^{-n}$, one step of amplitude amplification suffices. Coherent fanout circuits achieve the same constant-depth bound, while circuits using only one- and two-qubit gates attain depth $\Theta(n)$ with $O(4^n\log^2(n+2))$ gates. Generic targets require $\Omega(4^n/(h+1))$ qubits at depth $h$, even under unrestricted classical control. Our depth-width trade-offs meet this bound within a factor $O(n^2)$ for $1\le h\le2^n$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Mid-circuit measurements with feedforward reduce the optimal worst-case quantum depth for exact $n$-qubit unitary synthesis from $\Theta(n)$ to $O(1)$, using arbitrary one- and two-qubit gates and unrestricted ancillary space.	result 0.88	0.80	0.93	asserted 1.00
s1	Measurements and resets without feedforward remain subject to the circuit causal-cone bound.	result 0.41	0.24	0.03	asserted 1.00
s2	Our deterministic construction uses single-qubit measurements, $O(1)$ rounds of parity feedforward, and $O(n4^n\log(n+2))$ total qubits, with all ancillas starting and ending in $\|0\rangle$.	method 0.79	0.95	0.13	asserted 1.00
s3	Classical processing is not counted in quantum depth.	background 0.35	0.22	0.03	asserted 1.00
s4	We teleport only the matrix error of an approximation by Nehoran and Yuen.	method 0.95	0.82	0.03	asserted 0.99
s5	An exactly prepared ancillary state encodes this error, whose action is added back by coherent interference.	method 0.99	0.75	0.03	asserted 1.00
s6	At operator-norm error $2^{-n}$, one step of amplitude amplification suffices.	method 0.78	0.68	0.03	asserted 0.99
s7	Coherent fanout circuits achieve the same constant-depth bound, while circuits using only one- and two-qubit gates attain depth $\Theta(n)$ with $O(4^n\log^2(n+2))$ gates.	result 0.97	0.75	0.09	asserted 1.00
s8	Generic targets require $\Omega(4^n/(h+1))$ qubits at depth $h$, even under unrestricted classical control.	background 0.51	0.29	0.04	asserted 1.00
s9	Our depth-width trade-offs meet this bound within a factor $O(n^2)$ for $1\le h\le2^n$.	result 0.93	0.89	0.08	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Residual repair. (a) Schematic picture of the blocks. Direct Choi encoding (dotted route)

**Figure 2.** Schematic depth–width landscape for generic unitaries, with constants and logarithmic factors

Figures reproduced from arXiv:2610.06551 by the authors.

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Ben Foxman, Natalie Parham, Francisca Vasconcelos, and Henry Yuen. “Random unitaries in constant (quantum) time”. In Shubhangi Saraf, editor, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Volume 362 of Leibniz International Proceedings in Informatics (LIPIcs), pages 61:1–61:25. Dagstuhl, Ger- many(2026).SchlossDagstuhl–Leibniz-Z...

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Elisa Bäumer and Stefan Woerner. “Measurement-based long-range entangling gates in constant depth”. Phys. Rev. Research7, 023120 (2025). arXiv:2408.03064

View on Pith Pith review arXiv 2025
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View on Pith Pith review arXiv 2004
[40]

Gregory Rosenthal. “Bounds on theQAC 0 complexity of approximating parity”. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Volume 185 of LIPIcs, pages 32:1–32:20. (2021). arXiv:2008.07470

View on Pith Pith review arXiv 2021
[41]

Daniel Grier, Jackson Morris, and Kewen Wu. “QAC0 containsTC 0 (with many copies of the input)” (2026). arXiv:2601.03243v1

View on Pith arXiv 2026
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Maosen Fang, Stephen Fenner, Frederic Green, Steven Homer, and Yong Zhang. “Quantum lower bounds for fanout”. Quantum Inf. Comput.6, 46–57 (2006). arXiv:quant-ph/0312208

View on Pith Pith review arXiv 2006
[43]

Shivam Nadimpalli, Natalie Parham, Francisca Vasconcelos, and Henry Yuen. “On the Pauli spectrum ofQAC0”. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC). Pages 1498–1506. (2024). arXiv:2311.09631

View on Pith Pith review arXiv 2024
[44]

Anurag Anshu, Yangjing Dong, Fengning Ou, and Penghui Yao. “On the computa- tional power ofQAC0 with barely superlinear ancillae”. In Proceedings of the 57th An- nual ACM Symposium on Theory of Computing (STOC). Pages 1476–1487. (2025). arXiv:2410.06499

View on Pith arXiv 2025
[45]

Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. “Fast and efficient exact synthesis of single-qubit unitaries generated by Clifford and T gates”. Quantum Inf. Comput.13, 607–630 (2013). arXiv:1206.5236

View on Pith Pith review arXiv 2013
[46]

Brett Giles and Peter Selinger. “Exact synthesis of multiqubit Clifford+T circuits”. Phys. Rev. A87, 032332 (2013). arXiv:1212.0506

View on Pith Pith review arXiv 2013
[47]

Yasuhiro Takahashi and Seiichiro Tani. “Collapse of the hierarchy of constant-depth exactquantumcircuits”. Comput.Complexity25, 849–881(2016). arXiv:1112.6063v2

View on Pith Pith review arXiv 2016
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Shmuel Friedland, Lek-Heng Lim, and Jinjie Zhang. “An elementary and unified proof of Grothendieck’s inequality”. Enseign. Math.64, 327–351 (2018). arXiv:1711.10595. 12 Supplementary Material A Gaussian approximate synthesis We use Nehoran and Yuen’s continuous identity [21, Theorem 3.1], which they relate to Fiurášek’s continuous-variable gate synthesis ...

View on Pith Pith review arXiv 2018
[50]

If Pv|0⟩=|v,0⟩, applyP † v first, reflect about the zero state of the mode andallits prepara- tion work, and then applyPv. The resulting operator is Pv(I−2|0⟩⟨0|)P † v =I−2|v,0⟩⟨v,0|.(S15) Thus the reflection preserves the zero-work subspace. The preparation usess=O(q2K) qubits by Lemma 4. Compiling the reflection on these qubits costsO(slog(s+ 1)) = O(q3...

View on Pith

Discussion

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View on Pith Pith review arXiv 2025

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View on Pith Pith review arXiv 2004

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View on Pith Pith review arXiv 2021

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Daniel Grier, Jackson Morris, and Kewen Wu. “QAC0 containsTC 0 (with many copies of the input)” (2026). arXiv:2601.03243v1

View on Pith arXiv 2026

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Maosen Fang, Stephen Fenner, Frederic Green, Steven Homer, and Yong Zhang. “Quantum lower bounds for fanout”. Quantum Inf. Comput.6, 46–57 (2006). arXiv:quant-ph/0312208

View on Pith Pith review arXiv 2006

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Shivam Nadimpalli, Natalie Parham, Francisca Vasconcelos, and Henry Yuen. “On the Pauli spectrum ofQAC0”. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC). Pages 1498–1506. (2024). arXiv:2311.09631

View on Pith Pith review arXiv 2024

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View on Pith arXiv 2025

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View on Pith Pith review arXiv 2013

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Brett Giles and Peter Selinger. “Exact synthesis of multiqubit Clifford+T circuits”. Phys. Rev. A87, 032332 (2013). arXiv:1212.0506

View on Pith Pith review arXiv 2013

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Yasuhiro Takahashi and Seiichiro Tani. “Collapse of the hierarchy of constant-depth exactquantumcircuits”. Comput.Complexity25, 849–881(2016). arXiv:1112.6063v2

View on Pith Pith review arXiv 2016

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View on Pith 1979

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Shmuel Friedland, Lek-Heng Lim, and Jinjie Zhang. “An elementary and unified proof of Grothendieck’s inequality”. Enseign. Math.64, 327–351 (2018). arXiv:1711.10595. 12 Supplementary Material A Gaussian approximate synthesis We use Nehoran and Yuen’s continuous identity [21, Theorem 3.1], which they relate to Fiurášek’s continuous-variable gate synthesis ...

View on Pith Pith review arXiv 2018

[50] [50]

If Pv|0⟩=|v,0⟩, applyP † v first, reflect about the zero state of the mode andallits prepara- tion work, and then applyPv. The resulting operator is Pv(I−2|0⟩⟨0|)P † v =I−2|v,0⟩⟨v,0|.(S15) Thus the reflection preserves the zero-work subspace. The preparation usess=O(q2K) qubits by Lemma 4. Compiling the reflection on these qubits costsO(slog(s+ 1)) = O(q3...

View on Pith