arXiv:2610.03659

Quantum Codeword Sensing

John Blue, Zachary E. Chin, Vlad S. Oros, Isaac L. Chuang

quant-phSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations

Abstract

Consider the task of identifying an unknown classical codeword delivered to a register of quantum sensor qubits through a sequence of weak phase interactions that each encode one bit. Suppose that each received bit controls an identical $Z$-rotation of angle $θ$ applied to every sensor qubit; receivers may also use auxiliary qubits within a total qubit budget $Q$, with arbitrary control unitaries and entanglement among all qubits allowed during each reception step. When the total collected signal is weak ($Qθ\rightarrow 0$), we establish that a sensor coupled to a persistent quantum memory can identify the original codeword more reliably than any receiver with the same $Q$ and $θ$ but only classical memory between received bits, even when bit flips occur during transmission. In particular, for a family of $[n,k]$ binary simplex codes with even $k$ and $n=2^k-1$, we achieve exact correction of up to $t\leq(k-2)/4$ transmission errors by coupling the sensor register to a persistent quantum memory of the same size, using $Q = Θ((t+1)\log n)$ and $θ=\frac{2π}{k\sqrt{n+1}}$; for $t=o(\sqrt k)$, every comparable receiver with only classical memory instead has average success probability (over uniformly chosen codewords) tending to zero. Furthermore, under independent bit flips of any fixed probability $p<1/16$, our receiver with persistent quantum memory achieves decoding error tending to zero as $n\to\infty$ using $Q=O(\log n\log\log n)$ total qubits and the same $θ$. In contrast, every otherwise equivalent receiver with only classical memory has vanishing average success probability under the same stochastic error model.

Comments: 22 pages (13 figures)

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arXiv:2610.03659

Quantum Codeword Sensing

John Blue, Zachary E. Chin, Vlad S. Oros, Isaac L. Chuang

quant-phSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations

arXiv HTML PDF

Abstract

Z

-rotation of angle

θ

applied to every sensor qubit; receivers may also use auxiliary qubits within a total qubit budget

Q

, with arbitrary control unitaries and entanglement among all qubits allowed during each reception step. When the total collected signal is weak (

Qθ\rightarrow 0

), we establish that a sensor coupled to a persistent quantum memory can identify the original codeword more reliably than any receiver with the same

Q

and

θ

but only classical memory between received bits, even when bit flips occur during transmission. In particular, for a family of

[n,k]

binary simplex codes with even

k

and

n=2^k-1

, we achieve exact correction of up to

t\leq(k-2)/4

transmission errors by coupling the sensor register to a persistent quantum memory of the same size, using

Q = Θ((t+1)\log n)

and

θ=\frac{2π}{k\sqrt{n+1}}

; for

t=o(\sqrt k)

, every comparable receiver with only classical memory instead has average success probability (over uniformly chosen codewords) tending to zero. Furthermore, under independent bit flips of any fixed probability

p<1/16

, our receiver with persistent quantum memory achieves decoding error tending to zero as

n\to\infty

using

Q=O(\log n\log\log n)

total qubits and the same

θ

. In contrast, every otherwise equivalent receiver with only classical memory has vanishing average success probability under the same stochastic error model.

Comments: 22 pages (13 figures)

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Z

-rotation of angle

θ

applied to every sensor qubit; receivers may also use auxiliary qubits within a total qubit budget

Q

, with arbitrary control unitaries and entanglement among all qubits allowed during each reception step. When the total collected signal is weak (

Qθ\rightarrow 0

), we establish that a sensor coupled to a persistent quantum memory can identify the original codeword more reliably than any receiver with the same

Q

and

θ

but only classical memory between received bits, even when bit flips occur during transmission. In particular, for a family of

[n,k]

binary simplex codes with even

k

and

n=2^k-1

, we achieve exact correction of up to

t\leq(k-2)/4

transmission errors by coupling the sensor register to a persistent quantum memory of the same size, using

Q = Θ((t+1)\log n)

and

θ=\frac{2π}{k\sqrt{n+1}}

; for

t=o(\sqrt k)

p<1/16

, our receiver with persistent quantum memory achieves decoding error tending to zero as

n\to\infty

using

Q=O(\log n\log\log n)

total qubits and the same

θ

. In contrast, every otherwise equivalent receiver with only classical memory has vanishing average success probability under the same stochastic error model.

p

. For this noise structure, we demonstrate that, given a total sensing time

T

, Heisenberg scaling can be attained for times up to

T\propto 1/p^{(N+1)/2}

for a

N

-qubit repetition code, in contrast with

T\propto 1/p

without using a fault-tolerant sensing protocol.

m

bosonic modes. The null consists of all pure Gaussian states with total mean photon number at most one. Alternatives have half trace distance at least

1/32

from every pure Gaussian state; neither their energy nor the comparison family's energy is capped. Mixed states and infinite photon tails are included. Each visit accesses one entire original once. The total quantum dimension retained between visits is at most

D

, while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability

a

, or is erased with a classical flag. The quantity

k_a(D,m)

is the smallest deterministic original-preparation cap achieving worst-case error at most

1/3

. Failed preparations, discarded trials and reserved positions are all charged. A retained register of

q

qubits has

D=2^q

. An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is

\Theta(m)

. Along arbitrary growing-mode sequences, the precise criterion for improving that order is

k_a(D,m)=o(m)\quad\Longleftrightarrow\quad am\to\infty\ \text{and}\ \frac{aD}{m}\to\infty.

The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when

a=O(1/m)

, yet a survival probability tending to zero can still permit an advantage. Let

A=\max\{1,m^2/D\}

. At the fixed energy and separation above, constants

c,C>0

, independent of

m,D,a

, give

c\min\{m,A/a\}\le k_a(D,m)\le C\min\{m,A/a+\min(a^{-2},m^{2/3})\},\qquad 0<a\le1.

a=0

, the sample cost is

\Theta(m)

. For ideal memory the complete rate is

k_1(D,m)=\Theta\!\left(\min\{m,\max(1,m^2/D)\}\right).

With loss, the bounds match whenever

A\ge\min\{a^{-1},am^{2/3}\}

. For example,

a=m^{-1/6}

and

D=\lceil m^{11/6}\rceil

give

\Theta(m^{1/3})

preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for

D\ge m^2

a=m^{-1/2}

, the stated bounds are

\Omega(m^{1/2})

and

O(m^{2/3})

. Why the construction and converse fit the same resources A physical completely positive contraction combines first- and second-moment information before the next wait. Although the temporary pair output has order

m^2

labels, only a positive register of dimension

m+1

is retained. The three-visit route requires

D\ge m+1

and pays two actual survival events; a classical alternative handles the centering statistics. The converse covers arbitrary adaptive instruments and persistent noncommuting memory. Erasure removes the coherent prediction contribution while preserving information already learned in the full classical history. Unflagged noise and applicability For a fixed memoryless channel on the entire retained register with a physical reset decomposition

\mathcal N=a\Psi+(1-a)\mathcal R_\tau

, where

\Psi

is CPTP and

\mathcal R_\tau(X)=\operatorname{Tr}(X)\tau

, an exact flagged-erasure simulation gives

k_{\mathcal N}(D,m)\ge k_a(D,m)

. The established positive quantum Doeblin coefficient optimizes this lower obstruction. Whole-register depolarizing noise with unchanged fraction

1-q=O(1/m)

therefore forces

\Theta(m)

samples for every memory dimension. General unflagged noise inherits a converse, not the flagged model's complete achievability guarantees. The results quantify asymptotic preparation and persistent-memory resources under ideal visit-local control. Temporary auxiliary systems and gate complexity have no imposed budget, and the explicit sufficient finite-sample constants are conservative. The task is pure Gaussianity testing, rather than membership in the full family of mixed Gaussian states. The single integrated PDF contains the short main text, full proofs, four figures and precise resource accounting.