A Framework For Estimating Amplitudes of Quantum State With Single-Qubit Measurement

Nhat A. Nghiem

REVIEW 4 major objections 4 minor 22 references

A Framework For Estimating Amplitudes of Quantum State With Single-Qubit Measurement

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read The paper claims that measuring one qubit in arbitrary bases recovers every complex amplitude of an n-qubit pure state via a nonlinear equation system.

desk verdict The central claim fails: single-qubit measurement probabilities depend only on each qubit's reduced density matrix, so states like (|00>+|11>)/√2 and (|00>-|11>)/√2 are indistinguishable no matter how many angles are used. read the letter →

arxiv 2412.07123 v1 pith:JKQRTLF3 submitted 2024-12-10 quant-ph

Nhat A. Nghiem This is my paper · ORCID

classification quant-ph MSC 81P6881P15 PACS 03.65.Wj03.67.-a

keywords quantumstatetomographysingle-qubitmeasurementamplitudeestimationnonlinearalgebraicequationsBornrulesamplecomplexitybasispurerecovery

topics The Measurement Problem

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

The paper proposes that all complex amplitudes of an unknown $n$-qubit pure state $|\psi\rangle = \sum_{i=0}^{2^n-1} a_i |i\rangle$ can be recovered by measuring only one qubit at a time, as long as the measurement basis can be chosen freely. The outcomes are turned, through the Born rule, into a system of nonlinear algebraic equations whose unknowns are the real and imaginary parts of every amplitude, and solving that system classically yields the full state. The author derives sample counts for three error criteria: $O(4^n/\delta^4)$ single-qubit measurements for all amplitude norms to additive error $\delta$, $O(6^n/\delta^4)$ for total-variation distance $\delta$, and $O(2^n/\delta^4)$ for average $L_1$ error. If the framework holds, an arbitrary pure state would be fully characterizable from its single-qubit marginals alone, without ever measuring qubits jointly.

What carries the argument

The object that carries the argument is the system of nonlinear algebraic equations built from Born-rule probabilities of single-qubit measurements. A measurement of one qubit in the basis $\{|\theta\rangle_0, |\theta\rangle_1\}$, with $|\theta\rangle_0 = \cos\theta|0\rangle + \sin\theta|1\rangle$ and $|\theta\rangle_1 = \sin\theta|0\rangle - \cos\theta|1\rangle$, yields one equation per outcome expressing a sum of rotated-amplitude moduli squared as an estimable probability; the Z-basis measurements supply the marginal constraints. Together these give $2^{n+1}$ equations in the $2^{n+1}$ variables $\mathrm{Re}(a_i), \mathrm{Im}(a_i)$. The second load-bearing mechanism is Theorem 1, which bounds the shift of the solution when the right-hand sides are noisy, converting the per-probability estimation error $\epsilon$ into the sample-complexity claims through the Jacobian norm $\|J^{-1}\|$.

What would settle it

Measure the two-qubit states $(|00\rangle + |11\rangle)/\sqrt{2}$ and $(|00\rangle - |11\rangle)/\sqrt{2}$ through the paper's procedure: for either qubit and in every basis, the two outcome probabilities are $1/2$ and $1/2$ for both states, so the right-hand sides of the equation system coincide exactly even though the amplitude $a_{11}$ differs in sign. Writing out the equations for these two states and observing that the systems are identical would settle whether single-qubit outcomes can separate them.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that measurement outcomes from a single qubit, taken in the computational (Z) basis on every qubit and then in $M = \lceil 2^n - n \rceil$ additional rotated bases on the first qubit, determine every amplitude $a_i$, real and imaginary parts included. Each rotated-basis outcome contributes equations of the form $\sum_{i_1,...,i_{n-1}} |\cos\theta\, a_{0i_1...i_{n-1}} \pm \sin\theta\, a_{1i_1...i_{n-1}}|^2 = p^{(\theta)}_{0/1}$, and the Z-basis outcomes contribute marginal equations; together they form a square system of $2^{n+1}$ equations in the $2^{n+1}$ real unknowns. The paper proves an error-propagation bound (Theorem 1): when the right-hand sides are estimated from finitely many measurements, the solution error satisfies $|\tilde{x} - x| \le \|J^{-1}\|\,|\tilde{b} - b|$, with $J$ the Jacobian of the system. Assuming $\|J^{-1}\| = O(1)$, it then derives the three stated measurement counts, and concludes that joint multi-qubit measurement is not asymptotically stronger than single-qubit measurement for this task.

Load-bearing premise

The framework assumes that the probabilities obtained by measuring a single qubit in arbitrary bases carry enough information to uniquely determine all complex amplitudes, so that the nonlinear system has the true amplitudes as its only solution.

Editorial extensions

If this is right

If the framework is sound, a pure state can be characterized without ever measuring two qubits jointly, relaxing the hardware requirements of conventional multi-qubit tomography.
The claimed counts set the trade-offs: additive amplitude error costs $O(4^n/\delta^4)$, total-variation error $O(6^n/\delta^4)$, and average $L_1$ error $O(2^n/\delta^4)$, which for the last metric matches the naive all-qubit baseline under the $\|J^{-1}\| = O(1)$ condition stated in the paper.
The construction makes the choice of measurement basis the source of extra information, so the framework frames basis selection, not joint measurement, as the resource that yields full state recovery.
The paper draws the corollary that joint measurements are not asymptotically stronger than single-qubit ones for amplitude estimation, since any joint measurement would merely generate a different set of equations of the same kind.

Reading between the lines

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

The uniqueness of the solution is an implicit assumption the paper does not examine: two two-qubit states with identical single-qubit marginals, such as $(|00\rangle + |11\rangle)/\sqrt{2}$ and $(|00\rangle - |11\rangle)/\sqrt{2}$, produce identical right-hand sides for every single-qubit basis, so any recovery claim must confront this degeneracy.
All three sample-complexity formulas carry a hidden factor $\|J^{-1}\|^2$; the paper concedes it cannot prove $\|J^{-1}\| = O(1)$ and expects the behavior to be revealed numerically, so the advertised counts are conditional on that norm staying bounded.
A natural extension would be to count classical operations as well: solving $2^{n+1}$ coupled nonlinear equations may dominate the quantum measurement cost, and choosing the angles $\theta$ to minimize the Jacobian condition number is an unexplored optimization problem.
The framework raises an information-theoretic question it does not answer: which discrete sets of single-qubit bases are sufficient, and minimal, for a pure state to be determined by its marginals, since the choice $M = \lceil 2^n - n \rceil$ is proposed but not proven minimal.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Candidly: this paper is not close to correct. The central claim—that arbitrary single-qubit measurements can determine all amplitudes of an n-qubit pure state—is false, and the paper's own equations show why. Every probability in Eq. (71) is of the form Tr(|θ><θ|⊗I_{≠k} |ψ><ψ|), which depends only on the one-qubit reduced density matrix of the measured qubit. For the two-qubit states |ψ_+>=(|00>+|11>)/√2 and |ψ_->=(|00>-|11>)/√2, all single-qubit reduced states are I/2, so every single-qubit outcome probability is 1/2 for every basis and every angle. The full equation system is identical for these two states, so no solution procedure can tell them apart. The paper claims to recover the complex amplitudes in the Z basis; that is information-theoretically impossible from these marginals.

What is genuinely new here is modest: the explicit angle-dependent equations for single-qubit probabilities are not in the cited tomography papers, and the idea of building a nonlinear system from them is at least clearly presented. The author also deserves credit for stating up front that the Jacobian norm |J^{-1}| is unproven, and for noting that the approach would need it to be O(1). That honesty does not rescue the argument; it just identifies one more unsupported step.

The soft spots beyond the central impossibility are numerous and serious. The equation count ignores that p0+p1=1 makes half of the probabilities redundant, so the system has far fewer independent equations than variables. Theorem 1's proof is a linearization that drops higher-order terms and then states a bound as if it were exact; it also assumes the Jacobian is invertible at the solution, which fails for states with degenerate marginals. The sample-complexity claims all carry the unproven |J^{-1}| factor, so the displayed O(4^n/δ^4), O(6^n/δ^4), and O(2^n/δ^4) scalings are conditional on the very assumption that is most in doubt.

This is not a paper for a serious referee. It is a worked example of why single-qubit marginals are insufficient for full state reconstruction, though the author does not draw that conclusion. A desk reject is appropriate; if the author wants to pursue the idea, the correction is to acknowledge that only the reduced density matrices can be estimated, and the nonlinear system is overdetermined for that smaller goal.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a framework to estimate all amplitudes of an n-qubit pure state from measurements of a single qubit at a time, in the computational basis and in M = ceil(2^n - n) arbitrary rotated bases. The measurement probabilities are assembled into a system of nonlinear algebraic equations (Eq. 71), and the author argues that solving this system yields approximations to all real and imaginary parts of the amplitudes. The paper derives sample complexity bounds for three error metrics: O(4^n/δ^4) for maximum absolute norm error, O(6^n/δ^4) for total variation, and O(2^n/δ^4) for average L1 error, all under the assumption that the inverse Jacobian norm |J^{-1}| is O(1).

Significance. If the central claim were true, it would imply that full pure-state tomography is possible from single-qubit marginals, which would be a striking result. The manuscript is clearly written in parts and explicitly states its main assumption about the Jacobian. However, the claim fails on information-theoretic grounds: every probability obtained from a single-qubit measurement is a function only of the measured qubit's reduced density matrix, so the equations in Eq. (71) cannot distinguish states with identical single-qubit reduced states. The numerical complexity claims are therefore not established, and the proposed method cannot work as stated.

major comments (4)

[II.A, Eq. (71); Algorithm 1] The central claim is information-theoretically impossible. For any single-qubit measurement on qubit k in basis {|θ_0>,|θ_1>}, the outcome probability is Tr((|θ_s><θ_s| ⊗ I_{≠k})ρ) = <θ_s|ρ_k|θ_s>, where ρ_k is the reduced density matrix of qubit k. Hence every right-hand side in Eq. (71) depends only on the n single-qubit reduced density matrices, i.e., on at most 3n real parameters, whereas a pure n-qubit state has 2^{n+1}-2 real parameters. Concretely, for n=2 the states (|00>+|11>)/√2 and (|00>-|11>)/√2 have identical reduced density matrices (each equal to I/2), so every single-qubit measurement outcome probability is 1/2 in every basis; the equation systems (Eq. 7 or Eq. 71) are identical for the two states and cannot lead to different amplitude estimates. This contradicts the abstract's claim that arbitrary-basis single-qubit outcomes 'can be used to assist the finding of amplitudes.' It also implies that the Jacobian of the system at these states is singular, so the invertibility premise of Theorem 1 fails exactly in this regime.
[II.B, Eq. (71); Algorithm 1] The equation count is off by a factor of two. Each single-qubit measurement basis yields two probabilities p0 and p1 with p0+p1=1, so only one independent equation per basis. With n Z-basis measurements and M = ceil(2^n - n) rotated-basis measurements, Algorithm 1 produces at most n + M = 2^n independent equations, not the 2^{n+1} equations needed to determine the 2^{n+1} real variables. The displayed system (Eq. 71) lists 2n + 2M equations, but half are redundant. For the n=2 example, Eq. (7) has four rows but only three independent rows (since p0^1+p1^1=1 and p0^2+p1^2=1), and the claim of a 'square linear system' in the 3-qubit case (Eq. 26) similarly includes redundant rows. The system is therefore underdetermined by a factor of two, and no unique solution can be guaranteed.
[II.B, Theorem 1 and Eq. (90)] The perturbation bound in Theorem 1 is not valid as stated. The proof replaces f_i(˜x) by its first-order Taylor expansion around x and then treats J(˜x - x) = ˜b - b as an exact identity. Taylor's theorem only guarantees f_i(˜x) = f_i(x) + ∇f_i(x)·(˜x-x) + O(|˜x-x|^2), so the omitted remainder can be arbitrarily large unless a uniform bound on the second derivatives is established. No such bound is given, and the statement |˜x - x| ≤ |J^{-1}| |˜b - b| does not follow. Because all three sample-complexity claims (Eqs. (122), the total-variation bound in II.B.2, and the average-L1 bound in II.B.3) rely on Eq. (90), this is a load-bearing gap in the derivation.
[III. Conclusion] The manuscript explicitly acknowledges in the conclusion that 'we are not able to prove' that |J^{-1}| = O(1) and that the Jacobian behavior 'can only be revealed numerically.' Since the headline measurement counts O(4^n/δ^4), O(6^n/δ^4), and O(2^n/δ^4) are all derived after dropping the |J^{-1}| factor, these complexity claims are conditional on an unproven and, as shown in the first major comment, sometimes false assumption. For states with all single-qubit marginals equal to I/2, the Jacobian is singular, so |J^{-1}| is infinite and the bounds provide no guarantee. The complexity results are therefore not established.

minor comments (4)

[Throughout (e.g., after Eq. (70) and II.B)] The text uses 'complex variables' where 'real variables' is meant: a 2^n-dimensional complex amplitude vector has 2^{n+1} real degrees of freedom, not 2^{n+1} complex variables; this should be corrected.
[Algorithm 1] Algorithm 1 says to 'measure the first qubit O(2^n - n) times'; this should read 'measure the first qubit in O(2^n - n) different bases' or 'for O(2^n - n) values of θ', because copying the same measurement setting is not what is intended.
[II.B (Theorem 1 heading) and II.B (after Eq. (7))] There are several typographical errors, e.g., 'Algerbraic' in the Theorem 1 heading and 'the back equation 7, 14' in Section II.B; these should be corrected.
[II.A and Eq. (71)] The notation for probabilities is inconsistent: p0/p1 is reused for the first and second qubit in Section II.A (leading to the superscript notation introduced there), but the superscript is omitted in Eq. (71) for the Z-basis probabilities. Clarifying this would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper's central flaw is non-injectivity of single-qubit marginals, not a definitional or fitted-input loop.

full rationale

The paper does not fit parameters to data and does not rely on load-bearing self-citations or imported uniqueness theorems. Its derivation chain is self-contained but invalid for a non-circular reason. Eq. (27) defines each single-qubit measurement probability as Tr(|θ⟩⟨θ| ⊗ I_{n−1} · |ψ⟩⟨ψ|), which equals ⟨θ|ρ_k|θ⟩ for the reduced density matrix of the measured qubit. Consequently, every right-hand side in the nonlinear system (71) depends only on the n single-qubit reduced density matrices. For n = 2, (|00⟩+|11⟩)/√2 and (|00⟩−|11⟩)/√2 have identical reduced density matrices (I/2 for each qubit), so all equations in (71) are identical; no solution can distinguish the states. This is an information-theoretic impossibility, not a circular equivalence. Separately, Section III explicitly concedes that |J^{-1}| = O(1) is unproved: "We expect such a norm to grow as much as O(1), however, we are not able to prove it." The abstract's O(4^n/δ^4), O(6^n/δ^4), and O(2^n/δ^4) bounds drop this factor from Eq. (90). That is a serious unsupported assumption and should be weighed as a correctness risk, but it is not a fitted input or a self-citation chain. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No fitted physical constants and no new physical entities are introduced. The hidden assumptions are identifiability and conditioning: that single-qubit measurements determine all amplitudes and that the resulting nonlinear system has a well-behaved inverse Jacobian. Both are unsupported.

free parameters (2)

M = ceil(2^n - n)
The number of additional single-qubit measurement bases on the first qubit is hand-chosen so that the paper can claim 2^{n+1} equations. The count ignores that each basis provides only one independent equation.
Measurement angles theta_1,...,theta_M = unspecified
The protocol requires choosing these angles, yet no criterion or analysis of their effect on the Jacobian is given. The claimed |J^{-1}|=O(1) must hold for the chosen angles, which is never established.

assumptions (4)

standard math Born's rule probabilities for single-qubit measurements are as written in Eq. (70).
This is standard quantum mechanics and is not the issue.
ad hoc to paper The nonlinear system Eq. (71) has a unique solution equal to the true amplitudes.
This is the central false assumption. Single-qubit marginals do not determine the joint state, so the system cannot be uniquely solvable in general.
ad hoc to paper The Jacobian of Eq. (71) is invertible with ||J^{-1}||=O(1).
The paper states this cannot be proven and expects numerical evidence. All sample complexity claims depend on this unproven condition.
ad hoc to paper First-order Taylor expansion gives an exact error bound for the perturbed nonlinear system.
Theorem 1 drops higher-order terms without justification, so the bound |x~ - x| <= |J^{-1}| |b~ - b| is not rigorous for nonlinear equations.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Framework For Estimating Amplitudes of Quantum State With Single-Qubit Measurement." pith.science (2026). https://pith.science/paper/JKQRTLF3

@misc{pith2026241207123,
  author       = {Pith},
  title        = {Pith review of: A Framework For Estimating Amplitudes of Quantum State With Single-Qubit Measurement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JKQRTLF3}},
  note         = {Machine review of arXiv:2412.07123}
}

abstract

We propose and analyze a simple framework for estimating the amplitudes of a given $n$-qubit quantum state $\ket{\psi} = \sum_{i=0}^{2^n-1} a_i \ket{i}$ in computational basis, utilizing a single-qubit measurement only. Previously, it was a common procedure that one could measure all qubits in order to collect measurement outcomes, from which one can estimate amplitudes of given quantum state. Here, we show that if restricting to single-qubit measurement, and one can perform measurement on arbitrary basis, then the measurement outcomes can be used to assist the finding of amplitudes in the usual computational, or Z basis. More concretely, such outcomes are capable of constructing a system of nonlinear algebraic equations, and by classically solving them, we obtain $\Tilde{a}_i$, which is the approximation to the corresponding amplitudes $a_i$, including both real and imaginary component. We then discuss our framework from a broader perspective. First, we show that estimating all (norms of) amplitudes to additive accuracy $\delta$, i.e., $| |\Tilde{a}_i - |a_i| | \leq \delta$ for all $i$, $\mathcal{O}(4^n/\delta^4)$ single-qubit measurements is sufficient. Second, we show that to achieve total variation $\sum_{i=0}^{2^n-1} | |\Tilde{a}_i|^2 - |a_i|^2| \leq \delta $, $\mathcal{O}(6^n/\delta^4)$ a single bit measurement is required. Finally, in order to achieve an average $L_1$ norm error $ \sum_{i=0}^{2^n-1} | |\Tilde{a}_i| - |a_i| |/2^n \leq \delta$, a single bit measurement $\mathcal{O}(2^n/ \delta^4)$ is needed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 13 canonical work pages

[1]

This means that the right-hand side of the above linear system is erroneous

Then we also have that, for second qubit: p00 + p10 = p2 0 (5) p01 + p11 = p2 1 (6) It is straightforward to see that the above equation form a linear system:   1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1     p00 p01 p10 p11   =   p1 0 p1 1 p2 0 p2 1   (7) As we have discussed, O(ln(1/δ)/ϵ2) repetition is required to estimate p1 0, p2 0 and p1 1, p2...

work page
[2]

Recall that |ψ⟩2 = a000 |000⟩ + a001 |001⟩ + a010 |010⟩ + a011 |011⟩ + a100 |100⟩ + a101 |101⟩ + a110 |110⟩ + a111 |111⟩ (8) where each amplitude squared |aijk |2 = pijk is the probability of obtaining corresponding outcome ( i, j, k). Then due to marginal probability property, we have that: p000 + p001 + p010 + p011 = p1 0 (9) p100 + p101 + p110 + p111 =...

work page
[3]

Estimating each amplitude and its norm with additive errorδ Suppose our n-qubit state is |ψ⟩ = P i0,i1,...,in−1=0,1 ai0i1...in−1 |i0i1...in−1⟩ where each ai0i1...in−1 is a complex number, and we are interested in estimating its norm |ai0i1...in−1 | to additive error ϵ, for all amplitudes. As it is a complex number, we have that: |ai0i1...in−1 | = q ℜ(ai0i...

work page
[4]

Estimating probability distribution with total variationδ Suppose our n-qubit state is |ψ⟩ = P i0,i1,...,in−1=0,1 ai0i1...in−1 |i0i1...in−1⟩, if we measure all qubits in computational basis, then apparently the outcome i with corresponding probability p(i0i1...in−1) = |ai0i1...in−1 |2 forms a probability distribution, denoted as p. If we wish to estimate ...

work page
[5]

Estimating with average L1-norm accuracy Again, let |ψ⟩ = P i0,i1,...,in−1=0,1 ai0i1...in−1 |i0i1...in−1⟩ and instead of individual error δ as in section II B 1, we expect the average error to be δ: X i0,i1,...,in−1 1 2n ||˜ai0i1...in−1 | − |ai0i1...in−1 || ≤δ (147) Similar as before, let ai0i1...in−1 = ℜ(ai0i1...in−1 ) + iℑ(ai0i1...in−1 ), and ˜ai0i1...i...

work page
[6]

Simulating physics with computers

Richard P Feynman. Simulating physics with computers. In Feynman and computation, pages 133–153. CRC Press, 2018

2018
[7]

Quantum theory, the church–turing principle and the universal quantum computer

David Deutsch. Quantum theory, the church–turing principle and the universal quantum computer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 400(1818):97–117, 1985

1985
[8]

Rapid solution of problems by quantum computation

David Deutsch and Richard Jozsa. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439(1907):553–558, 1992

1907

Show all 22 references

[9]

Universal quantum simulators

Seth Lloyd. Universal quantum simulators. Science, 273(5278):1073–1078, 1996

1996
[10]

Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer

Peter W Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM review, 41(2):303–332, 1999

1999
[11]

A fast quantum mechanical algorithm for database search

Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219, 1996

1996
[12]

Predicting many properties of a quantum system from very few measurements

Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16(10):1050–1057, 2020

2020
[13]

Efficient estimation of pauli observables by derandomization.Physical review letters, 127(3):030503, 2021

Hsin-Yuan Huang, Richard Kueng, and John Preskill. Efficient estimation of pauli observables by derandomization.Physical review letters, 127(3):030503, 2021

2021
[14]

Sample efficient tomography via pauli measurements

Nengkun Yu. Sample efficient tomography via pauli measurements. arXiv preprint arXiv:2009.04610, 2020

2009 arXiv
[15]

Collective versus local measurements in a qubit mixed-state estimation

E Bagan, M Baig, Ramon Mu˜ noz-Tapia, and A Rodriguez. Collective versus local measurements in a qubit mixed-state estimation. Physical Review A, 69(1):010304, 2004

2004
[16]

Quantum state estimation and large deviations

Michael Keyl. Quantum state estimation and large deviations. Reviews in Mathematical Physics, 18(01):19–60, 2006

2006
[17]

Optimal estimation of qubit states with continuous time measurements

M˘ ad˘ alin Gut ¸˘ a, Bas Janssens, and Jonas Kahn. Optimal estimation of qubit states with continuous time measurements. Communications in Mathematical Physics, 277:127–160, 2008

2008
[18]

Sample-optimal tomography of quantum states

Jeongwan Haah, Aram W Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. Sample-optimal tomography of quantum states. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 913–925, 2016

2016
[19]

Efficient quantum tomography

Ryan O’Donnell and John Wright. Efficient quantum tomography. In Proceedings of the forty-eighth annual ACM sympo- sium on Theory of Computing, pages 899–912, 2016

2016
[20]

Fast state tomography with optimal error bounds.Journal of Physics A: Mathematical and Theoretical, 53(20):204001, 2020

Madalin Gut ¸˘ a, Jonas Kahn, Richard Kueng, and Joel A Tropp. Fast state tomography with optimal error bounds.Journal of Physics A: Mathematical and Theoretical, 53(20):204001, 2020

2020
[21]

Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators

Steven T Flammia, David Gross, Yi-Kai Liu, and Jens Eisert. Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. New Journal of Physics, 14(9):095022, 2012

2012
[22]

Low rank matrix recovery from rank one measurements

Richard Kueng, Holger Rauhut, and Ulrich Terstiege. Low rank matrix recovery from rank one measurements. Applied and Computational Harmonic Analysis, 42(1):88–116, 2017

2017

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

This means that the right-hand side of the above linear system is erroneous

Then we also have that, for second qubit: p00 + p10 = p2 0 (5) p01 + p11 = p2 1 (6) It is straightforward to see that the above equation form a linear system:   1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1     p00 p01 p10 p11   =   p1 0 p1 1 p2 0 p2 1   (7) As we have discussed, O(ln(1/δ)/ϵ2) repetition is required to estimate p1 0, p2 0 and p1 1, p2...

work page

[2] [2]

Recall that |ψ⟩2 = a000 |000⟩ + a001 |001⟩ + a010 |010⟩ + a011 |011⟩ + a100 |100⟩ + a101 |101⟩ + a110 |110⟩ + a111 |111⟩ (8) where each amplitude squared |aijk |2 = pijk is the probability of obtaining corresponding outcome ( i, j, k). Then due to marginal probability property, we have that: p000 + p001 + p010 + p011 = p1 0 (9) p100 + p101 + p110 + p111 =...

work page

[3] [3]

Estimating each amplitude and its norm with additive errorδ Suppose our n-qubit state is |ψ⟩ = P i0,i1,...,in−1=0,1 ai0i1...in−1 |i0i1...in−1⟩ where each ai0i1...in−1 is a complex number, and we are interested in estimating its norm |ai0i1...in−1 | to additive error ϵ, for all amplitudes. As it is a complex number, we have that: |ai0i1...in−1 | = q ℜ(ai0i...

work page

[4] [4]

Estimating probability distribution with total variationδ Suppose our n-qubit state is |ψ⟩ = P i0,i1,...,in−1=0,1 ai0i1...in−1 |i0i1...in−1⟩, if we measure all qubits in computational basis, then apparently the outcome i with corresponding probability p(i0i1...in−1) = |ai0i1...in−1 |2 forms a probability distribution, denoted as p. If we wish to estimate ...

work page

[5] [5]

Estimating with average L1-norm accuracy Again, let |ψ⟩ = P i0,i1,...,in−1=0,1 ai0i1...in−1 |i0i1...in−1⟩ and instead of individual error δ as in section II B 1, we expect the average error to be δ: X i0,i1,...,in−1 1 2n ||˜ai0i1...in−1 | − |ai0i1...in−1 || ≤δ (147) Similar as before, let ai0i1...in−1 = ℜ(ai0i1...in−1 ) + iℑ(ai0i1...in−1 ), and ˜ai0i1...i...

work page

[6] [6]

Simulating physics with computers

Richard P Feynman. Simulating physics with computers. In Feynman and computation, pages 133–153. CRC Press, 2018

2018

[7] [7]

Quantum theory, the church–turing principle and the universal quantum computer

David Deutsch. Quantum theory, the church–turing principle and the universal quantum computer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 400(1818):97–117, 1985

1985

[8] [8]

Rapid solution of problems by quantum computation

David Deutsch and Richard Jozsa. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439(1907):553–558, 1992

1907