Displaced-Parity Sampling

• Displaced-Parity Sampling is a dual-purpose framework that uses parity-only observation in distribution testing and displaced photon-number parity in phase-space sampling for quantum optics.
• In the distribution testing context, it converts label misalignment into observable run statistics, leading to tight sample complexity bounds blending classical and parity-trace bottlenecks.
• In quantum optics, it underpins displaced detection strategies for boson sampling and cat state certification, linking experimental methods with computational hardness.

Displaced-Parity Sampling denotes several distinct constructions in contemporary theory and quantum optics. In the literature represented here, it refers, first, to parity-only observation under systematic label misalignment in distribution testing, where an ordered sample reveals only the least significant bit of each draw and nearby labels may be “displaced” or confused on a path or cycle; and, second, to phase-space sampling of the displaced photon-number parity operator in linear optics and cat-state certification, where parity is measured after a displacement and thereby samples the Wigner function at chosen phase-space points (Jr. et al., 2023, Seshadreesan et al., 2014, Matulík et al., 21 Jun 2026).

1. Scope and basic formalism

In the parity-trace setting, the basic observable is the parity function

$b(i)= i \bmod 2,$

applied to an ordered i.i.d. sample $S=(X_1,\dots,X_m)$ from an unknown distribution over $[n]$. The tester does not observe the labels $X_t$ themselves; it observes only the binary sequence $Y_t=b(X_t)$, while preserving order. The associated distance notion is total variation,

$\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$

Over $[2n]$, the analysis separates odd and even parts and studies the resulting run structure of the binary trace (Jr. et al., 2023).

In the quantum-optical setting, the core operators are the displacement operator and the parity operator,

$$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$

with $\hat n=a^\dagger a$. The displaced-parity operator is

$\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$

For a single mode, the Wigner function is linked to displaced parity by

$S=(X_1,\dots,X_m)$0

while the catability work adopts the convention

$S=(X_1,\dots,X_m)$1

In both conventions, expectation values of displaced parity sample the Wigner function at the chosen phase-space point (Seshadreesan et al., 2014, Matulík et al., 21 Jun 2026).

The displacement in these usages is therefore not the same object. In (Jr. et al., 2023), displacement is a mislabeling or local confusability abstraction on a path or cycle. In (Seshadreesan et al., 2014) and (Matulík et al., 21 Jun 2026), displacement is the phase-space operator $S=(X_1,\dots,X_m)$2 acting on bosonic modes.

2. Parity-trace distribution testing and the displaced-label abstraction

The formal testing problem asks whether an unknown distribution $S=(X_1,\dots,X_m)$3 is uniform, given only the parity sequence $S=(X_1,\dots,X_m)$4: decide whether $S=(X_1,\dots,X_m)$5 versus $S=(X_1,\dots,X_m)$6 with success probability at least $S=(X_1,\dots,X_m)$7, using as few samples as possible. Much of the analysis is performed over $S=(X_1,\dots,X_m)$8, writing the odd and even partial distributions as

$S=(X_1,\dots,X_m)$9

so that the observed trace is a binary string whose alternating runs of $[n]$0s and $[n]$1s encode information about $[n]$2 through run lengths (Jr. et al., 2023).

The displaced-label interpretation is formalized by the confused collector model. Here the base graph $[n]$3 is a path or cycle on $[n]$4 vertices, and a resolution parameter $[n]$5 controls how accurately adjacent vertices are separated. Each edge is included with probability $[n]$6 to form a random subgraph $[n]$7. One then draws $[n]$8 i.i.d. samples from $[n]$9 over $X_t$0, but each sample is labeled only by the representative of its connected component in $X_t$1. Samples from indistinguishable joined vertices therefore collapse to the same label. As $X_t$2 decreases, nearby labels are more often confused (Jr. et al., 2023).

The parity trace over $X_t$3 becomes a weighted confused collector on a cycle of odd positions. Edges between adjacent odd vertices are included with probability

$X_t$4

namely the event that no even index between them appears in the trace. The resulting connected components are the observable buckets, and parity runs are precisely these connected components. This means that the observed order carries coarse adjacency information while suppressing absolute position. A plausible implication is that order becomes the critical surrogate for identity once labels are systematically misaligned.

3. Tight uniformity testing under the parity trace

The main technical theorem gives a tight sample complexity for parity-trace uniformity testing:

$X_t$5

The $X_t$6 term is the classical $X_t$7/collision term from fully labeled uniformity testing, while the new $X_t$8 term is the parity-trace bottleneck. The crossover occurs at $X_t$9: for $Y_t=b(X_t)$0, $Y_t=b(X_t)$1 dominates, whereas for $Y_t=b(X_t)$2, $Y_t=b(X_t)$3 dominates (Jr. et al., 2023).

The upper-bound algorithm has three steps. The first is a bias test: count the total number $Y_t=b(X_t)$4 of $Y_t=b(X_t)$5s, and reject if $Y_t=b(X_t)$6 differs significantly from $Y_t=b(X_t)$7, with threshold $Y_t=b(X_t)$8. The second is a concentration test on the circularized trace: compute the maximum run length among $Y_t=b(X_t)$9s and $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$0s, and reject if the maximum run length exceeds $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$1. The third is a collision-based run statistic. If $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$2 are the positive run lengths of $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$3s in the circular trace, define

$\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$4

where $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$5 is the hidden odd-count vector and $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$6 is the join matrix indicating whether odd indices lie in the same connected component. The tester rejects when $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$7 exceeds the expected uniform baseline by at least $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$8 (Jr. et al., 2023).

The analysis proceeds through expectation separation and variance control. With $\|p-q\|_{\mathrm{TV}}=\frac12\sum_i |p(i)-q(i)|.$9, one has

$[2n]$0

On a path, $[2n]$1 is Toeplitz; on a cycle, it is circulant. Writing $[2n]$2, where $[2n]$3 is the “uniform conjugate” of $[2n]$4, gives

$[2n]$5

so the separation from uniformity is driven by $[2n]$6. The associated concentration bound in the parity-trace model is

$[2n]$7

The approximate uniform conjugate is built so that each connected component induced by $[2n]$8 has approximately equal expected odd mass $[2n]$9, with

$$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$0

This reduction makes the spectral term $$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$1 central to both the mean and variance analyses (Jr. et al., 2023).

The lower bound is based on a domino construction. The domain $$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$2 is partitioned into $$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$3 consecutive dominoes of length $$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$4, each covering $$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$5. The even parts are fixed as $$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$6. In the YES case, each domino is uniform. In the NO case, each domino independently receives one of the two local biases

$$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$7

with equal probability. Dominoes with at most two samples reveal no information about the $$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$8 sign; only $$D(\alpha)=\exp(\alpha a^\dagger-\alpha^* a), \qquad \Pi = (-1)^{\hat n},$$9-way or higher collisions matter. The resulting mutual-information bound is

$\hat n=a^\dagger a$0

which, via Fano’s inequality, yields

$\hat n=a^\dagger a$1

in the regime $\hat n=a^\dagger a$2. Combined with the standard fully labeled lower bound $\hat n=a^\dagger a$3, this gives the full lower bound

$\hat n=a^\dagger a$4

4. Consequences for confused collectors, property testing, and trace reconstruction

The parity-trace abstraction yields direct results for the confused collector on cycles and paths. For $\hat n=a^\dagger a$5 and $\hat n=a^\dagger a$6, testing uniformity on $\hat n=a^\dagger a$7 requires

$\hat n=a^\dagger a$8

This interpolates between the standard $\hat n=a^\dagger a$9 rate at $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$0 and $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$1 when $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$2 is as small as the theorem allows. The displacement here arises because connected components join contiguous vertices, so the tester must exploit ordering and collision-like statistics to undo local confusability (Jr. et al., 2023).

The same framework yields equivalences between parity-trace distribution testing and distribution-free sample-based property testing for density properties. A labeled distribution $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$3 induces an alternation sequence and a density sequence $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$4; membership in a density property depends only on that sequence. The equivalence is stated using an edit distance on fractional strings of the form

$\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$5

where insert/delete zero-length blocks and rearrangement of adjacent equal blocks are free, while changing block length by $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$6 costs $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$7. Two distributions are indistinguishable under parity trace iff this edit distance is $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$8. Within this framework, testing support size $\Pi(\alpha)=D(\alpha)\Pi D^\dagger(\alpha).$9 under parity trace matches distribution-free sample-based testing of $S=(X_1,\dots,X_m)$00-alternating functions, giving the first tight equivalence between distribution testing and property testing for a non-symmetric property. The promised uniformly $S=(X_1,\dots,X_m)$01-alternating problem has sample complexity

$S=(X_1,\dots,X_m)$02

and the reductions recover lower bounds $S=(X_1,\dots,X_m)$03 for $S=(X_1,\dots,X_m)$04-alternating functions on $S=(X_1,\dots,X_m)$05 and $S=(X_1,\dots,X_m)$06 for halfspaces on $S=(X_1,\dots,X_m)$07 (Jr. et al., 2023).

There is also a black-box reduction to property testing in trace reconstruction. If $S=(X_1,\dots,X_m)$08 denotes the deletion channel on $S=(X_1,\dots,X_m)$09 with retention rate $S=(X_1,\dots,X_m)$10, then a single trace can be “Poissonized” to a parity trace of $S=(X_1,\dots,X_m)$11, where

$S=(X_1,\dots,X_m)$12

and $S=(X_1,\dots,X_m)$13 matches a $S=(X_1,\dots,X_m)$14 sample. This yields, among other consequences, a one-trace tester for whether $S=(X_1,\dots,X_m)$15 is a uniform $S=(X_1,\dots,X_m)$16-block string versus $S=(X_1,\dots,X_m)$17-far in relative edit distance, with expected trace size

$S=(X_1,\dots,X_m)$18

and a tester for whether $S=(X_1,\dots,X_m)$19 is an $S=(X_1,\dots,X_m)$20-block string versus $S=(X_1,\dots,X_m)$21-far with expected trace size $S=(X_1,\dots,X_m)$22. A lower bound gives $S=(X_1,\dots,X_m)$23 for any $S=(X_1,\dots,X_m)$24-trace tester. The multiple-trace uniform-$S=(X_1,\dots,X_m)$25-block problem admits a matching lower bound

$S=(X_1,\dots,X_m)$26

This suggests that the parity-trace abstraction functions as a transport mechanism between distribution testing, sample-based property testing, and trace reconstruction rather than as an isolated model.

5. Linear-optical displaced parity and computational complexity

In linear optics, displaced-parity sampling arises from displaced photon-number detection. At the outputs of an interferometer, one applies mode-wise displacements $S=(X_1,\dots,X_m)$27, implementable by unbalanced homodyne, and then performs number-resolving detection. From the resulting distribution $S=(X_1,\dots,X_m)$28, the parity statistic is obtained by coarse-graining modulo $S=(X_1,\dots,X_m)$29:

$S=(X_1,\dots,X_m)$30

equivalently

$S=(X_1,\dots,X_m)$31

Since displaced parity is a linear functional of displaced number sampling, it does not introduce additional post-processing complexity beyond number sampling. In boson sampling’s no-collision regime $S=(X_1,\dots,X_m)$32, where each mode contains at most one photon with high probability, per-mode parity is identical to on-off detection (Seshadreesan et al., 2014).

For displaced single-photon Fock states (DSPFS), the single-mode input is $S=(X_1,\dots,X_m)$33, and the $S=(X_1,\dots,X_m)$34-mode input is

$S=(X_1,\dots,X_m)$35

Under a passive linear-optical interferometer $S=(X_1,\dots,X_m)$36,

$S=(X_1,\dots,X_m)$37

Applying the counter-displacement $S=(X_1,\dots,X_m)$38 exactly removes the coherent offsets and yields the usual AA output superposition with amplitudes proportional to matrix permanents,

$S=(X_1,\dots,X_m)$39

Consequently, sampling with DSPFS inputs and displaced detection is in the same complexity class as AA boson sampling for all displacement values. In the no-collision regime, per-mode parity after cancellation samples the same bitstrings as standard boson sampling. Total parity, by contrast, is fixed to $S=(X_1,\dots,X_m)$40 and is therefore uninformative (Seshadreesan et al., 2014).

For single-photon-added coherent states (SPACS), the normalized single-mode state is

$S=(X_1,\dots,X_m)$41

With equal amplitudes $S=(X_1,\dots,X_m)$42, output cancellation leads to a superposition over total photon-number sectors $S=(X_1,\dots,X_m)$43, with probabilities

$S=(X_1,\dots,X_m)$44

This produces a computational-complexity transition. If $S=(X_1,\dots,X_m)$45, then $S=(X_1,\dots,X_m)$46 remains inverse-polynomial, and at $S=(X_1,\dots,X_m)$47,

$S=(X_1,\dots,X_m)$48

If $S=(X_1,\dots,X_m)$49, then $S=(X_1,\dots,X_m)$50, so the AA term dominates. By contrast, when $S=(X_1,\dots,X_m)$51 scales as $S=(X_1,\dots,X_m)$52 or larger, the vacuum sector dominates,

$S=(X_1,\dots,X_m)$53

and the output becomes classically simulatable. The same transition carries over to displaced parity sampling: in the small-$S=(X_1,\dots,X_m)$54 regime, post-selection on the $S=(X_1,\dots,X_m)$55-photon sector retains AA hardness, whereas in the large-$S=(X_1,\dots,X_m)$56 regime displaced-parity expectations become trivial and the output is Gaussian-like (Seshadreesan et al., 2014).

6. Parity-less Kerr cats and generalized catability

In the certification of parity-less Kerr cat states, displaced-parity sampling is used because Kerr cats cannot be distinguished by parity at the origin. The construction begins with the nullifier

$S=(X_1,\dots,X_m)$57

whose nullspace is spanned by $S=(X_1,\dots,X_m)$58 and $S=(X_1,\dots,X_m)$59. Earlier catability used

$S=(X_1,\dots,X_m)$60

but for Kerr cats the parity term is replaced by displaced parity,

$S=(X_1,\dots,X_m)$61

The resulting approximate nullifier is

$S=(X_1,\dots,X_m)$62

Here the displacement $S=(X_1,\dots,X_m)$63 is chosen to target the shifted interference fringes characteristic of parity-less Kerr cats (Matulík et al., 21 Jun 2026).

For equatorial cat states

$S=(X_1,\dots,X_m)$64

the displaced-parity expectation used for $S=(X_1,\dots,X_m)$65-selection is

$S=(X_1,\dots,X_m)$66

The paper gives

$S=(X_1,\dots,X_m)$67

with

$S=(X_1,\dots,X_m)$68

Under $S=(X_1,\dots,X_m)$69 and $S=(X_1,\dots,X_m)$70, one obtains the simplified forms quoted in the source, and for Kerr cats $S=(X_1,\dots,X_m)$71,

$S=(X_1,\dots,X_m)$72

This makes $S=(X_1,\dots,X_m)$73 a direct control parameter for locating the central positive or negative fringe (Matulík et al., 21 Jun 2026).

Because $S=(X_1,\dots,X_m)$74 for general $S=(X_1,\dots,X_m)$75, the paper introduces a normalized nullifier

$S=(X_1,\dots,X_m)$76

where

$S=(X_1,\dots,X_m)$77

This normalization yields expectation values that approach zero on ideal Kerr cats. The work also identifies an exact anti-linear nullifier based on complex conjugation $S=(X_1,\dots,X_m)$78, with

$S=(X_1,\dots,X_m)$79

and

$S=(X_1,\dots,X_m)$80

These nullify ideal Kerr cats in expectation but are not directly measurable, so displaced parity serves as the practical surrogate (Matulík et al., 21 Jun 2026).

The measurement protocol evaluates $S=(X_1,\dots,X_m)$81 from photon-number statistics at a finite set of displacements. Writing

$S=(X_1,\dots,X_m)$82

one has the estimator

$S=(X_1,\dots,X_m)$83

and the source gives an explicit decomposition of $S=(X_1,\dots,X_m)$84 and of $S=(X_1,\dots,X_m)$85 in terms of $S=(X_1,\dots,X_m)$86, $S=(X_1,\dots,X_m)$87, $S=(X_1,\dots,X_m)$88, and $S=(X_1,\dots,X_m)$89. The displacement settings $S=(X_1,\dots,X_m)$90 therefore suffice. The generalized catability is

$S=(X_1,\dots,X_m)$91

with $S=(X_1,\dots,X_m)$92 witnessing cat-like non-Gaussianity. Loss is modeled by the pure-loss channel with Kraus operators

$S=(X_1,\dots,X_m)$93

The numerical benchmarks use $S=(X_1,\dots,X_m)$94, optimize $S=(X_1,\dots,X_m)$95, set $S=(X_1,\dots,X_m)$96, and report that the generalized catability is more resilient to optical loss than fidelity-based certification, especially for smaller $S=(X_1,\dots,X_m)$97 (Matulík et al., 21 Jun 2026).

7. Distinctions, limitations, and open directions

The main distinction across these usages is operational. In distribution testing, displaced-parity sampling is fundamentally a model of partial observability with preserved order: the tester sees only parity bits and run lengths, and the difficulty comes from systematic misalignment of nearby labels. The cited work explicitly relies on i.i.d. sampling, on order preservation, and on path or cycle structure; it identifies extensions to trees, identity and closeness testing under parity trace, multi-bit traces, and other channels such as insertions and substitutions as open problems (Jr. et al., 2023).

In linear optics, the principal distinction is between total parity and per-mode parity. Total parity is often too coarse: it is fixed for DSPFS and becomes trivial in the large-$S=(X_1,\dots,X_m)$98 SPACS regime. Per-mode parity in the no-collision regime, by contrast, coincides with on-off detection and therefore preserves the hardness of AA boson sampling after displacement cancellation. This suggests that the computational content of displaced-parity sampling depends sharply on whether parity is used as a global scalar observable or as a mode-resolved bitstring (Seshadreesan et al., 2014).

In Kerr-cat certification, the generalized nullifier with displaced parity is approximate when written as $S=(X_1,\dots,X_m)$99, because $[n]$00; the normalized version $[n]$01 corrects this by inserting the ideal target value $[n]$02. The same work emphasizes further limitations: accurate phase-stable calibration of $[n]$03 is essential, finite-sample statistics and detector nonidealities introduce variance and bias, the anti-linear nullifier based on $[n]$04 is not directly measurable, and explicit sample-complexity bounds are not provided. A plausible implication is that displaced-parity sampling is most powerful when it targets a small number of physically motivated phase-space points rather than attempting full tomography (Matulík et al., 21 Jun 2026).