New Frontiers in Quantum-Enhanced Sensing and Metrology · NFQESM 2026

Quantum Error Detection for Heisenberg-Limited Sensing

Quantum sensing viewed as computation with an unknown gate.

Carlos Ortiz Marrero · Rui Jie Tang · Nathan Wiebe

Pacific Northwest National Laboratory and Colorado State University

NFQESM 2026 · Durham · September 17, 2026

Pacific Northwest National Laboratory Colorado State University
One experiment, two descriptions

Ramsey sensing as a computational problem

Problem

Estimate signal phase \(\varphi\).

Sensing

  1. Prepare coherence.
  2. Expose: relative phase \(2\varphi\).
  3. Analyze and read populations.

Computing

\[p(Z=+1\mid\varphi)=\cos^2\varphi\]

Input: signal access

\(U(\varphi)=e^{-i\varphi Z}\)

\(\varphi\) unknown

Chosen controls

Preparation
and analysis

Output

Repeat and estimate \(\hat\varphi\)

Precision \(\epsilon\)
Confidence \(1-\delta\)

Cost

\(Q\) signal calls

One fixed-duration
exposure per probe
= one query

Design choice: organizing signal calls

Coherent signal calls amplify phase dependence

Design choice

Parallel: \(N\) entangled probes. Sequential: \(N\) exposures.

Parallel example: \(N=3\)

Parity readout

Measure \(X\) on each probe.

\[b=\prod_{k=1}^N x_k,\quad x_k=\pm1\]

Cost per shot

\(N\) signal calls

Signal dependence

\[p(b=+1\mid\varphi)=\frac{1+\cos 2N\varphi}2\]

Ideal GHZ parity statistics. Ortiz Marrero, Tang, Wiebe: GHZ sensing section.

After coherent accumulation: choosing the readout

Readout controls make ambiguous phases distinguishable

Problem

Coherent gain still leaves ambiguous phases.

Control: choose the analysis-pulse phase

Highlighted gates: known reference \(a\).
Same parity readout: \(b=x_1x_2x_3\).

Original: a=0Chosen: a=π/240φAφBπ/301/210.850.15φ candidate signal phasep(b=+1 | φ, a)

Original readout

\(a=0\)

\(p_A=1/2\)

\(p_B=1/2\)

Chosen readout

\(a=\pi/24\)

\(p_A\approx0.85\)

\(p_B\approx0.15\)

\(p_A=\Pr(b=+1\mid\varphi_A,a)\), \(p_B=\Pr(b=+1\mid\varphi_B,a)\): outcome probabilities for the two candidate phases.

Two candidate values become two candidate intervals

A promised task: distinguish interval A from interval B

Given

The phase lies in \(A\) or \(B\).

0π/6π/4π/3π/200.51ATransition gapBφ unknown phasep(X=+1 | φ, controls)Desired probability of +1

Task

Distinguish \(A\) from \(B\) with high probability.

Desired measurement

\(\varphi\in A\): \(X=+1\) likely.

\(\varphi\in B\): \(X=-1\) likely.

Gap

Outside the assumed inputs.

Allows a smooth probability transition.

Which controls implement this desired behavior?

Quantum signal processing (QSP)

QSP constructs a measurement for the A-versus-B task

Query budget
\(d\) signal calls

0π/6π/4π/3π/200.51ATransition gapBφ unknown phasep(X=+1 | φ, controls)Desired probability of +15-call QSP
  1. Choose the desired outcome probabilities over \(A\) and \(B\).
  2. Compute control angles that approximate this behavior.
  3. Keep those angles fixed during sensing.

Desired behavior: \(p_+ \equiv \Pr(X=+1\mid\varphi,\boldsymbol\theta)\)

\(\varphi\in A:\ p_+\approx1\qquad\varphi\in B:\ p_+\approx0\)

QSP theorem (informal)

QSP shapes quantum amplitudes.

An admissible degree-\(d\) amplitude polynomial can be implemented using \(d\) signal calls.

Degree, parity and unitarity conditions.

QSP theorem: Martyn et al. (2021), Theorem 1. Ideal five-call illustration: \(\boldsymbol\theta=(0,3\pi/8,0,\pi/4)\).

From one probe with zero rotations to \(N\) probes in parallel

GHZ implements a simple QSP response in parallel

Parallel: \(N\) probes, one call each

\(N=3\) shown. Parity \(b=x_1x_2x_3\).

Sequential: \(N\) calls, zero QSP rotations

\(\Delta=\varphi-a\)

\[A_N(\Delta)=\cos(N\Delta)=T_N(\cos\Delta)\]

\(T_N\): degree-\(N\) Chebyshev polynomial.

Same outcome probability, now for parity \(b\)

\[p_+\equiv\Pr(b=+1\mid\varphi,a)=|A_N(\Delta)|^2\]
\[p_+=\frac{1+\cos(2N\Delta)}{2}\]

The shape is fixed. Offset \(a\) shifts it.

Paper: GHZ State Implementation. QSP: Martyn et al. (2021), Theorem 1, zero-control specialization.

A quantum probability becomes a classical decision

Majority vote distinguishes separated phase regions

Local interval: \(|\varphi-\mu|\le\pi/(4N)\). Choose a boundary \(\mu\).

−π/4−π/120π/12π/401/41/23/41LeftGapRightN(φ − μ) offset from the decision boundaryp₊ = Pr(b = +1 | φ, a)

Noiseless curve shown.

  1. Set \(a=\mu+\pi/(4N)\).
  2. Collect \(m\) independent parity outcomes.
  3. Majority \(+1\): right.
    Majority \(-1\): left.

Per-decision scaling with noise

Gap half-width \(\epsilon=\pi/(12N)\), failure \(\le\delta\).

\(Q=Nm=O\!\left(\frac{e^{4N\sigma^2}}{\epsilon}\log\frac{1}{\delta}\right)\)

\(N\sigma^2\ll1\): inverse-resolution query scaling for this local decision.

Transition gap: \(|\varphi-\mu|<\pi/(12N)\). No guaranteed side.

Noise: fluctuating \(Z\) fields. \(U_k=e^{-i(\varphi+\xi_k)Z},\quad \xi_k\sim\mathcal N(0,\sigma^2)\).

Offsets are independent across probes and redrawn each shot. \(\sigma\): phase standard deviation.

Paper: Binary Decision Rule, Majority Vote Correctness and Noise Marginalization. Shown gap chosen for this illustration.

What changes with noise

Transverse noise changes the outcome probabilities

Problem

Now change the noise model: transverse \(X\) noise mixes error sectors.

\[V(\varphi,\gamma)=e^{-i(\varphi Z+\gamma X)}\]

Unknown signal

Phase \(\varphi\)

Calibrated transverse field

Transverse angle \(\gamma\)

Ideal operations

Preparation, controls,
checks and readout

Parity mixes error sectors. Record a sector label too.

Worked model: homogeneous transverse X noise, ideal preparation, controls and readout. Paper: bit-flip protocol.

A logical probe and an additional measurement

Checks label error sectors while preserving within-sector coherence

\[|0_L\rangle=|000\rangle,\quad |1_L\rangle=|111\rangle,\quad |+_L\rangle=|\mathrm{GHZ}_3\rangle.\]
Compatible basis statesSyndrome \(s\)
\(|000\rangle,\ |111\rangle\)\((+1,+1)\)
\(|010\rangle,\ |101\rangle\)\((-1,-1)\)

The same syndrome labels both branches. The checks commute with the signal and with \(X_1X_2X_3\) parity.

Repetition-code checks \(s=(Z_1Z_2,Z_2Z_3)\). Paper: bit-flip protocol.

The same design problem on a logical qubit

EQSP combines signal processing with a syndrome record

Repeat modules,
then read logical \(X\).

\(s=(s_1,s_2)\):
syndrome

Encoded QSP (EQSP): process the signal on a logical qubit and retain the syndrome.

Use syndrome and final readout together in inference. No active recovery.

Paper: Encoded Quantum Signal Processing definition. Compatible checks preserve the relevant within-sector phase.

The quantum experiment supplies data for inference

Each shot returns two pieces of information

Syndrome \(s\)

Which error sector occurred?

Parity \(b\)

What did the phase-sensitive measurement report?

Joint likelihood

\[p(s,b\mid\varphi)\]

The syndrome tells us how to interpret the parity outcome.

Use both outputs to estimate the phase.

Keep flagged and unflagged shots.

Vary the interrogation length and readout setting; combine the resulting records.

Paper: syndrome-conditioned phase estimation and error-detected full-likelihood (ED-FL) estimator.

The theoretical guarantee · fixed code size

Full-record estimation has inverse-precision query scaling

Guarantee

Precision \(\epsilon\), confidence \(1-\delta\)

\[\Pr\!\left(|\hat\varphi-\varphi|\le\epsilon\right)\ge1-\delta\]
\[Q_{\mathrm{total}}=O\!\left(\frac{\log(1/(\epsilon\delta))\,\log(1/\delta)}{\epsilon}\right)\]

Heisenberg query scaling, up to logarithmic factors.

Assumptions

Fixed code size \(N\). Small transverse noise. Ideal operations.

Calibrated identifiable likelihood on a known phase interval.

\(M_j\): signal calls per probe in shot \(j\).
\(Q_{\mathrm{total}}=N\sum_jM_j\): all calls, including flagged shots.

Ortiz Marrero, Tang, Wiebe: “ED-FL inherits the rejection-sampling resource bound” lemma.

Takeaway

Sensing is algorithm design

Design the quantum experiment and the classical inference together.

Choose the controls, record the syndrome, and infer the phase from the full record.

QR code for the paper on arXiv
arxiv.org/abs/2603.22798
Ortiz Marrero, Tang, Wiebe
Backup: the worked GHZ example

Parity probabilities depend on the recorded syndrome

The GHZ probe is \(|+_L\rangle\). Read the checks, then parity.

π/8π/43π/8π/200.51s=(+1,+1)s=(−1,−1)φ signal phase per callp(b=+1 | s,φ)

Exact model, \(N=3,\ \gamma=0.15\). Two of the four syndrome sectors are shown.

The same parity bit implies different phase likelihoods for different syndromes.

Paper: exact syndrome-sector amplitudes and joint parity statistics. Exact analytic illustration.

Backup: the full-record inference equations

The estimator uses the syndrome and parity together

Observed data

\(\mathcal D\): every \((s_j,b_j)\), with chosen controls \(\mathcal C_j=(M_j,a_j)\).

Likelihood

\[p(s,b\mid\varphi)=p(s\mid\varphi)\,p(b\mid s,\varphi)\]
Which sector occurred?
\[p(s\mid\varphi)\]
What did its parity report?
\[p(b\mid s,\varphi)\]

Updated phase distribution

\[\pi(\varphi\mid\mathcal D)\propto\pi_0(\varphi)\prod_jp(s_j,b_j\mid\varphi,\mathcal C_j)\]

Keep flagged and unflagged shots. A posterior replaces the majority vote.

Paper: syndrome-conditioned phase estimation and error-detected full-likelihood (ED-FL) estimator.

Backup · the paper's estimation procedure

The complete sensing procedure is a phase-estimation algorithm

  1. 1 Choose

    Signal multiple \(M_j\)
    Readout phase offset \(a_j\)

  2. 2 Record

    Syndrome \(s_j\)
    Parity \(b_j\)

  3. 3 Infer

    Update phase distribution
    Resolve ambiguity

Output

Estimate \(\hat\varphi\)

Precision \(\epsilon\), confidence \(1-\delta\)

Total cost

\[Q_{\mathrm{total}}=N\sum_{j=1}^{S}M_j\]

All shots. Fresh fixed \(N\)-qubit block each shot.

Preparation, controls and readout lie outside \(Q\).

Paper: bit-flip estimation algorithm and ED-FL resource lemma. The resource proof uses randomized interrogation settings.

Backup · width and depth

Sequential sensing accumulates phase through repeated calls

\[p_+=\frac{1+\cos2M(\varphi-a)}2,\qquad \delta\varphi\simeq\frac{1}{2M\sqrt{nS}},\qquad Q=\sum_r nS_rM_r.\]

\(n\) separate probes, \(M\) calls each. \(S\) shots. \(a\) is the analysis offset.

Independent Gaussian Z offsets of variance \(\sigma^2\), fixed within each shot: near-ideal visibility requires \(M_*^2\sigma^2\ll1\), where \(M_*\) is the largest depth.

Paper: sequential binary search and its longitudinal-noise theorem. Displayed probabilities and sensitivity are the ideal local formulas.

Backup · additional noise assumptions

Longitudinal inhomogeneity imposes a separate condition

Independent Gaussian Z offsets, redrawn each shot, with phase variance \(\sigma^2\):

\[p_+=\frac{1+e^{-2N\sigma^2}\cos2N(\varphi-a)}2.\]

Near-ideal GHZ visibility requires \(N\sigma^2\ll1\). The repetition-code checks do not identify these phase offsets.

The paper's combined construction uses a GHZ state of repetition-code blocks. Its guarantee also requires small transverse noise and \(N_{\mathrm{total}}\sigma^2=o(1)\).

Paper: Noise Marginalization lemma and Combined Protocol with Logical GHZ States theorem. These conditions are additional to the main example.

Backup · exact three-qubit response

The conditional curves retain both compatible patterns

\[\Omega=\sqrt{\varphi^2+\gamma^2},\quad f=\frac{\sin\Omega}{\Omega},\quad u=\cos\Omega-i\varphi f,\quad v=-i\gamma f.\]
\[c_x=\frac{u^{3-w}v^w+(u^*)^w v^{3-w}}{\sqrt2},\qquad w=|x|.\]

For a sector containing \(x,\bar x\), set \(c=c_x\), \(d=c_{\bar x}\). Parity exchanges these basis states.

\[p(s)=|c|^2+|d|^2,\qquad p(s,b)=\frac{|c+bd|^2}{2},\qquad p(b\mid s)=\frac{p(s,b)}{p(s)}.\]

At \(\gamma=0\), only \(s=(+1,+1)\) occurs and \(p(b=+1)=[1+\cos6\varphi]/2\). Use \(f=1\) at \(\Omega=0\).

Paper: exact syndrome-sector amplitudes and homogeneous joint statistics. Conditional probabilities require \(p(s)>0\).

Backup · scope of the resource lemma

The resource guarantee holds the code block fixed

Fixed odd \(N\ge3\), calibrated transverse X noise, no Y or longitudinal component. Ideal preparation, controls, syndrome extraction and readout.

\[\max_k|\gamma_k/\varphi|\ll1,\qquad Np_{\max}<1,\qquad \delta\in(0,1/2].\]

\(p_{\max}\) bounds per-qubit flip weight across interrogation settings. Use a compact identifiable phase interval, parameter-independent likelihood support, differentiability in quadratic mean, uniformly finite Fisher information and bounded score.

\[T_{\mathrm{total}}=\sum_jM_j,\qquad Q_{\mathrm{total}}=NT_{\mathrm{total}}.\]

The main \(O(\cdot)\) absorbs fixed code and noise parameters. It is not a uniform guarantee as code size or noise varies.

Paper: Maximum Likelihood Sampling Complexity and ED-FL inheritance lemmas, plus the bit-flip operating regime.