---
title: Fock-State Boson Sampling
url: https://www.emergentmind.com/topics/fock-state-boson-sampling
type: topic
---

# Fock-State Boson Sampling

Searching arXiv for recent and foundational papers on Fock-state boson sampling and related generalizations.
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arxiv_search.query({"search_query":"ti:\"BosonSampling with single-photon Fock states from a bright solid-state source\" OR id:1603.00054 OR ti:\"Boson sampling with displaced single-photon Fock states versus single-photon-added coherent states\" OR id:1402.0531 OR ti:\"Generalized concurrence in boson sampling\" OR id:1605.08506","start":0,"max_results":10,"sortBy":"relevance","sortOrder":"descending"})
arxiv_search.query({"search_query":"all:\"shallow-depth boson sampling\" OR all:\"generalized bosons\" OR all:\"higher spin sampling\" OR all:\"Entanglement in the full state vector of boson sampling\"","start":0,"max_results":10,"sortBy":"submittedDate","sortOrder":"descending"})
Fock-state boson sampling denotes the class of linear-optical sampling problems in which both the inputs and the readout are formulated in the Fock basis. In the canonical Aaronson–Arkhipov setting, \(n\) single-photon Fock states are injected into \(m\) modes of a passive interferometer, and one samples the resulting occupation-number distribution; in a broader usage, the term also includes arbitrary multimode Fock inputs with repeated occupations, for which the same permanent formalism survives with repeated rows and columns in the relevant submatrices [1402.0531][1605.08506]. The subject occupies a central position in quantum-optical complexity theory because the output amplitudes of noninteracting bosons are governed by matrix permanents, while experimentally it serves as a benchmark for how closely a physical platform can realize genuine bosonic many-body interference with fixed-number states rather than Gaussian or approximate surrogates [1603.00054].

## 1. Canonical formulation

The standard Fock-state boson-sampling model prepares
\[
|\psi_\mathrm{in}\rangle = |1_1,\dots,1_n,0_{n+1},\dots,0_m\rangle
= \left(\prod_{i=1}^{n}\hat{a}_i^\dagger\right) |0_1,\dots,0_m\rangle,
\]
evolves it through an \(m\times m\) passive linear-optical unitary \(\hat U\), and measures output occupations in the photon-number basis. The mode transformation is
\[
\hat{U}\hat{a}_i^\dagger\hat{U}^\dagger = \sum_{j=1}^m U_{i,j}\hat{a}_j^\dagger,
\]
so the output state is a superposition over all \(n\)-photon occupation strings,
\[
|\psi_\mathrm{out}^{\mathrm{AA}}\rangle = \sum_S \gamma_S \, |s_1^{(S)},\dots,s_m^{(S)}\rangle,
\qquad P(S)=|\gamma_S|^2.
\]
In the collision-free regime, each amplitude is proportional to a permanent,
\[
\gamma_S \propto \mathrm{Per}(U_{S,T}),
\]
which is the source of the model’s classical intractability conjecture [1402.0531].

The broader Fock-state formulation allows arbitrary occupation vectors
\[
|\vec n\rangle = |n_1,n_2,\ldots,n_M\rangle,\qquad \sum_{i=1}^M n_i=N,
\]
with possible repeated occupations in a mode. In that case the transition amplitude remains permanent-valued,
\[
\langle \vec{m}|\hat{U}|\vec{n}\rangle
=\frac{ \mathrm{Per}\big( [U]_{\vec{n},\vec{m}}\big)  }{\prod_{k=1}^{M}\sqrt{n_{k}! m_{k}!} },
\]
where \([U]_{\vec n,\vec m}\) is formed by repeating rows and columns according to input and output multiplicities [1605.08506]. In this sense, collision-free boson sampling is a sparse corner of a larger Fock-space scattering problem. A complementary formulation views the same amplitudes as overlaps between many-body Fock states defined with respect to two single-particle mode bases related by a linear canonical transformation, with the permanent emerging as the exact many-body scattering amplitude [1502.07483].

## 2. Why the Fock basis is the reference model

Fock-state boson sampling is not merely a historical baseline; it is the precise structure against which nearby input models are compared. A useful example is the contrast between displaced single-photon Fock states and single-photon-added coherent states. For displaced single-photon Fock states,
\[
\hat D(\alpha)=\exp\!\left(\alpha \hat a^\dagger-\alpha^*\hat a\right),\qquad
\hat D(\alpha)\hat a^\dagger|0\rangle,
\]
the displacement simply propagates through the interferometer and can be exactly undone by a counter-displacement before number detection, leaving the ordinary Aaronson–Arkhipov distribution. Accordingly, displaced single-photon Fock-state sampling with displaced photon-number detection is in the same complexity class as ordinary boson sampling for all displacements [1402.0531].

By contrast, single-photon-added coherent states,
\[
|\alpha,1\rangle \propto \hat a^\dagger |\alpha\rangle
= \hat a^\dagger \hat D(\alpha)|0\rangle,
\]
do not preserve the fixed-\(n\)-photon sector because \([\hat a^\dagger,\hat D(\alpha)] = \alpha^* \hat D(\alpha)\). After counter-displacement, the output contains a coherent superposition of sectors with total photon number from \(0\) to \(n\), and the \(n\)-photon boson-sampling sector occurs with probability
\[
P_n=\frac{1}{(1+|\alpha|^2)^n}.
\]
The model is “just as hard as AA boson sampling” when \(|\alpha|^2\le 1/n\), but becomes classically simulatable in the large-amplitude regime, specifically \(|\alpha|^2\ge n^2\), where the vacuum dominates [1402.0531]. This sharp transition isolates what is special about Fock inputs: they preserve the exact fixed-particle-number sector whose amplitudes are governed by permanents.

The same point clarifies the relation to Gaussian boson sampling. Gaussian boson sampling keeps photon counting at the output but replaces fixed Fock inputs by squeezed Gaussian states, changing the governing matrix function from the permanent to the hafnian [1612.01199]. Experimentally this improves generation rates, but it is a distinct model; it does not realize the original Fock-state input condition that underwrites the canonical boson-sampling formulation [1905.00170].

## 3. Genuine single-photon Fock-state implementations

A decisive experimental milestone was the realization of boson sampling with genuine single-photon Fock-state inputs from a bright solid-state source rather than approximate heralded photons from spontaneous parametric downconversion [1603.00054]. Earlier SPDC-based photonic implementations relied on the two-mode squeezed state
\[
\ket{\psi} = \sqrt{1-|\lambda|^2}\sum_{n=0}^{\infty}\lambda^n \ket{nn},
\]
with \(|\lambda|\ll 1\), so the single-photon sector was obtained only by heralding and strong pump suppression. The solid-state experiment replaced that approximation by an InGaAs quantum dot deterministically coupled to a micropillar cavity, operated at \(13\) K and excited quasi-resonantly at \(905\) nm with repetition rate \(R_L=80\) MHz, with emission at \(932\) nm collected into a single-mode fibre [1603.00054].

The source brightness followed
\[
\eta=\eta_0\left(1-e^{-P/P_0}\right),
\]
with \(\eta_0=0.14\) and \(P_0=150~\mu\text{W}\). At \(P=1.2P_0\), the source had single-photon purity
\[
1-g^{(2)}(0)=0.990\pm 0.001,
\qquad g^{(2)}(0)=0.010\pm0.001,
\]
and even at \(3P_0\) the purity remained \(0.976\pm0.001\) with
\[
g^{(2)}(0)=0.024\pm0.001.
\]
These numbers are the explicit experimental signature that the input states were very nearly \(|1\rangle\) Fock states rather than approximate pseudo-single photons [1603.00054].

To obtain a multiphoton input from one emitter, the experiment used passive temporal-to-spatial demultiplexing with chained beam splitters and \(12.5\) ns delay lines, routing three consecutive emissions into distinct spatial channels. The boson-sampling run injected \(N=3\) photons into input modes \(\{1,2,3\}\) of a \(6\times 6\) linear optical network \(\mathcal L\), realized with 3 spatial modes and 2 polarization-encoded modes. Because of losses, \(\mathcal L\) was treated as a generally non-unitary transfer matrix [1603.00054].

For perfectly indistinguishable photons, the three-photon probabilities obey
\[
p^{(3)}=\left|\mathrm{per}(\mathcal T)\right|^2,
\]
but the actual experiment operated in a partially distinguishable regime. The measured pairwise indistinguishabilities were
\[
\mathcal{I}_{1,2}=0.520,\qquad
\mathcal{I}_{2,3}=0.540,\qquad
\mathcal{I}_{1,3}=0.643,
\]
so the full three-photon theory contained permanent, determinant, and immanant contributions weighted by these overlaps. The measured three-photon distribution over all \(\binom{6}{3}=20\) no-collision outputs had fidelity
\[
\mathcal F=0.997\pm0.006,
\]
with \(6725\) three-fold events collected in \(9~\text{hours}\) [1603.00054].

The experiment also foregrounded efficiency. Using the \(n\)-photon probability per trial
\[
p_\text{pt}^{(n)}=\frac{c_\text{gen}^{(n)}}{R_\text{trial}}
=\frac{c_\text{det}^{(n)}}{\eta_d\,R_\text{trial}},
\]
it reported
\[
p_\text{pt}^{(3)}=2.8\times 10^{-5}\quad \text{at }1.2P_0,
\qquad
p_\text{pt}^{(3)}=7.1\times 10^{-5}\quad \text{at }3P_0,
\]
which the authors compared to prior SPDC-based values in the \(10^{-6}\) to \(10^{-7}\) range, concluding that the solid-state source was “between one and two orders-of-magnitude more efficient than current heralded multi-photon sources based on spontaneous parametric downconversion” [1603.00054]. The central significance of this experiment was therefore twofold: it realized boson sampling with genuine single-photon Fock inputs, and it materially improved multiphoton generation efficiency.

## 4. Repeated occupations and generalized Fock inputs

Although the canonical model uses one photon per occupied input mode, Fock-state boson sampling naturally extends to higher occupations. A minimal experimental probe of that regime is the interference of states
\[
|l\rangle_a |S-l\rangle_b,
\]
which includes genuine repeated occupations such as \(|2,2\rangle\), \(|0,4\rangle\), and \(|0,5\rangle\). In a two-mode beam splitter, the transition probabilities can be written as
\[
p(k) = \bigl\vert \langle k, S-k | U_{\text{BS}}^{(r)} | l, S-l\rangle\bigr\vert^2
      = \bigl\vert \phi_k^{(r)}(l-Sr,S) \bigr\vert^2,
\]
with Kravchuk functions replacing general permanent expressions in this analytically solvable \(2\)-mode setting [1906.00678]. This is not Aaronson–Arkhipov boson sampling, but it directly probes the repeated-occupation regime that a general Fock-state treatment must accommodate.

In the full many-mode setting, the exact permanent formula for repeated occupations leads to specialized exact algorithms. One notable result is a generalized permanent formula for \(\mathrm{Per}([A]_{\vec n,\vec m})\) together with a complexity measure, the Fock-state concurrence sum \(C_S\), for arbitrary input and output occupation patterns. For the exact algorithms analyzed, the minimal runtime is
\[
\mathcal T_{\min}(\vec{n},\vec{m})=
\mathcal O\!\left( 2^N \min[C_S(\vec n),C_S(\vec m)]\,\alpha_{\vec n}\alpha_{\vec m} \right),
\]
where \(\alpha_{\vec n}\) is the number of nonzero occupations in \(\vec n\). In this formulation, highly bunched inputs can be substantially easier than maximally spread collision-free inputs, and \(C_S\) acts as the collective occupation-pattern measure controlling exact runtime across the analyzed algorithms [1605.08506].

A further generalization replaces ordinary bosons by “generalized bosons,” for which off-site commutation remains bosonic but the local commutator becomes occupation dependent. Even then, the boson-sampling probabilities remain permanent governed:
\[
\Pr(\mathbf k|\mathbf l)
=
\left(\prod_i \frac{f(k_i)}{f(l_i)k_i!}\right)^2
\left|\mathrm{Perm}(\Lambda[\mathbf k|\mathbf l])\right|^2.
\]
This suggests that the permanent structure is tied more fundamentally to symmetric exchange across modes than to the strict local relation \([a_i,a_i^\dagger]=1\) [2204.08389].

## 5. Complexity, validation, and restricted architectures

The standard hardness narrative for Fock-state boson sampling is inherited from the permanent structure of its output amplitudes, but current work has focused on how much of that story survives under altered measurements, shallow depth, and noise. One example is boson sampling with Gaussian continuous-variable measurements: the input remains a Fock-state single-photon input \(\ket{1_N}\), the dynamics remain passive linear optics, and selected discretized CV events retain the same permanent core strongly enough to imply exact-sampling hardness, but the Aaronson–Arkhipov route to approximate-sampling hardness does not presently transfer because the hard events occupy too small a portion of outcome space [1705.06041].

Circuit depth is another fault line. For geometrically local shallow-depth linear-optical circuits, light-cone constraints make many outputs exactly impossible or effectively negligible. In a \(d\)-dimensional local parallel architecture, the light-cone size obeys
\[
|L_D(i)| \le \left(\frac{2D}{d}\right)^d,
\]
and the paper “Exploring Shallow-Depth Boson Sampling” shows that for sufficiently shallow depth most Fock-state boson-sampling outputs are exactly forbidden, obstructing anti-concentration and therefore the usual average-case-hardness route [2306.10671]. That work proposes a geometrically non-local shallow architecture, the non-local hypercubic structure, as a way to recover Haar-like statistics more rapidly.

A more recent complexity-theoretic development proves worst-case and average-case hardness of output-probability estimation for logarithmic-depth Fock-state boson sampling on the \((\mathcal B\mathcal B^*)^q\) architecture. In that setting the output law remains
\[
p_{\bm s}(C)=\frac{1}{\prod_{i=1}^{M}s_i!}\left|\mathrm{Per}(C_{\bm s,\bm t})\right|^2,
\]
and the paper establishes average-case \(\#\mathrm P\)-hardness for estimating \(p_{\bm s}(U)\) over random shallow circuits and random collision-free outputs, with a corresponding extension to a constant-rate local photon-loss model [2405.01786]. The authors are explicit, however, that this is not yet a full approximate-sampling hardness theorem; an additive-precision gap remains.

The complexity picture can also collapse in the opposite direction. In a model of non-unitary free-boson dynamics under repeated random forced measurement, the effective single-particle amplitudes localize, different bosons asymptotically evolve into the same few-site packet, and the output state approaches
\[
|\psi\rangle \approx (\hat b^\dagger)^N|0\rangle.
\]
In that frozen regime the output occupation distribution becomes simple, bunching dominated, and classically easy to sample. This establishes that the permanent-based description alone does not guarantee hardness once monitored dynamics drive the effective single-particle evolution into a localized low-support phase [2110.12230].

## 6. Characterization, state structure, and emerging extensions

A mature account of Fock-state boson sampling must include not only complexity and implementation, but also characterization of the resulting many-body state. A recent atomic boson-sampling characterization program measured indistinguishability using a Hong–Ou–Mandel-style experiment and reported
\[
99.5^{+0.5}_{-1.6}\%
\]
for indistinguishable atoms. It then connected that two-particle indistinguishability to multiparticle bunching features in Fock-basis output data, introduced the weak generalized bunching conjecture, and studied how to optimize experimental design for inferring the single-particle unitary from Fock-basis measurements; in that setting, “having very cold atoms was necessary to perform the inference of the dynamics in a reasonable amount of time” [2410.10593]. This line of work shifts the emphasis from mere sampling to certifying that a device is genuinely realizing the intended many-boson Fock-basis dynamics.

At the level of full-state structure, the output of ideal Fock-state boson sampling admits an exact generalized coherent-state expansion whose size is exponential only in photon number and polynomial in mode number. For the standard input of \(S\) single photons in the first \(S\) modes, the expansion contains \(2^{S-1}\) generalized coherent states, which enables exact calculations of Rényi entanglement entropies for moderate \(S\) and very large \(M\). The resulting mode entanglement exhibits symmetric Page-curve-like behavior, a maximum at equal partition, saturation with \(M\) in the collision-free regime, and a maximum entropy that scales linearly with \(S\) [2210.09915]. This suggests that the full boson-sampling state is far more structured than a list of output probabilities alone would indicate.

Alternative platforms have long been proposed. A trapped-ion architecture maps bosonic modes to local transverse phonon excitations, with deterministic preparation of phononic Fock states, near-unit-efficiency projective readout in the Fock basis, and universal linear bosonic mode mixing generated by Coulomb-induced phonon hopping plus fast single-mode phase shifts [1310.4860]. More speculative generalizations attempt to transplant Fock-state boson-sampling logic into non-photonic settings. “Beyond Boson Sampling: Higher Spin Sampling as a Practical Path to Quantum Supremacy” argues that a spin-\(S\) system can emulate an equivalent Fock-state boson-sampling task in a linear-mode regime with
\[
m\sim n^{1+\epsilon},\qquad \epsilon=\frac{3}{2S},
\]
interpolating from the spin-\(\tfrac12\) quartic regime toward quasi-linear scaling at larger \(S\) [2505.07312]. By contrast, the transmon-based proposal for “\(q\)-boson Fock state sampling” should not be identified with ordinary boson sampling: it is a \(q\)-deformed generalization whose ordinary-boson limit is \(q\to1\), and its complexity-theoretic status is not developed at the same level of precision as the permanent-based Aaronson–Arkhipov model [2506.21094].

Taken together, these developments define Fock-state boson sampling as more than the original three-line sampling problem. It is a family of permanent-governed many-boson interference tasks anchored by exact Fock-state inputs, extended to repeated occupations and generalized bosons, constrained in practice by indistinguishability, loss, and circuit architecture, and increasingly analyzed through state-space, validation, and platform-specific lenses rather than through output probabilities alone [1603.00054][2204.08389].

Source: https://www.emergentmind.com/topics/fock-state-boson-sampling