Bosonic Quantum Circuit Representation

• Bosonic quantum circuit representation is a framework that defines quantum circuits using bosonic modes, characterized by creation/annihilation operators and Fock space structure.
• It spans diverse implementations including linear-optical systems, continuous-variable qumodes, and encoded bosonic schemes on qubit hardware with both Gaussian and non-Gaussian gates.
• The approach integrates matrix functions like permanents and hafnians for computational complexity, and underpins fault-tolerant architectures through bosonic error correction codes.

Searching arXiv for recent and foundational papers on bosonic quantum circuit representations across linear optics, qumodes, hard-core bosons, and bosonic encodings. Search query: "bosonic quantum circuit representation linear optics qumodes hard-core bosons" Bosonic quantum circuit representation denotes a family of circuit formalisms in which bosonic modes, bosonic statistics, or bosonic encodings determine the state space, gate set, and measurement model. In the literature, the term covers passive linear-optical circuits lifted from mode unitaries to bosonic Fock space, qumode-based continuous-variable circuits in cavity and trapped-ion platforms, algebraic hard-core-boson representations of multi-qubit circuits, and qubitized discretizations of bosonic fields or Gaussian bosonic evolutions on exponentially many modes (McCloud, 2022, Dutta et al., 2024, Dutta et al., 2024, Emmanuel-Costa et al., 26 Jun 2026, Macridin et al., 2018, Barthe et al., 2024). Across these settings, the central objects are creation and annihilation operators, number operators, symplectic or unitary mode transformations, and measurement maps such as photon-number, parity, homodyne, heterodyne, or overlap estimation.

1. Operator-theoretic foundations

A single bosonic mode is a quantum harmonic oscillator with Fock basis $\{|n\rangle\}_{n=0}^\infty$, ladder operators $a,a^\dagger$ obeying $[a,a^\dagger]=1$, and number operator $\hat{n}=a^\dagger a$. In the standard continuous-variable parametrization, the quadratures are

$$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$

with $[x,p]=i$, and multimode systems are described by the phase-space vector $R=(x_1,p_1,\ldots,x_N,p_N)^T$ (Dutta et al., 2024). In passive linear optics, the same physics can be expressed in terms of a mode-mixing unitary $U$ whose lift to bosonic Fock space preserves total photon number and acts block-diagonally on fixed-photon-number sectors (McCloud, 2022).

Paradigm	Degrees of freedom	Circuit description
Linear optics	Multimode bosonic Fock space	Mode unitary $U$ lifted to $B[U]$
Qumodes/CV	Harmonic oscillators	Gaussian and non-Gaussian gates
Encoded bosons	Finite grids, HCB sites, qubit blocks	QFT-based digitization or algebraic encodings

The literature does not use a single universal formalization. One recurring distinction is between particle-preserving representations, where circuits are built from beam splitters and phase shifters and conserve total occupation number, and general bosonic representations, where squeezing, Kerr, SNAP, echoed conditional displacement, or cubic-phase resources enlarge the accessible algebra. A second distinction is between native bosonic hardware and encoded bosons. In the former, oscillator modes are physical; in the latter, bosonic degrees of freedom are represented on qubits through finite truncations, digitized fields, or hard-core boson algebras. This multiplicity is not a contradiction but a feature of the subject: “bosonic quantum circuit representation” names a circuit-level semantics rather than a single hardware prescription.

2. Linear-optical circuits and bosonic interference

In linear optical quantum computing, a circuit in one time-bin is generated by an $a,a^\dagger$0 unitary matrix $a,a^\dagger$1 acting on $a,a^\dagger$2 modes, with annihilation and creation operators transforming linearly. The lifted unitary $a,a^\dagger$3 on multimode bosonic Fock space is fixed by its commutation relations with mode operators and vacuum normalization, and amplitudes in fixed-photon-number sectors are given by normalized permanents of submatrices of $a,a^\dagger$4 (McCloud, 2022). This is the canonical bosonic circuit representation of passive interferometers and supplies the operator semantics for beam splitters, phase shifters, boson sampling, and KLM-type constructions.

For two indistinguishable particles entering input modes $a,a^\dagger$5 of a linear network, the bosonic output amplitude at detectors $a,a^\dagger$6 is

$a,a^\dagger$7

with probability $a,a^\dagger$8. The corresponding antisymmetric amplitude is

$a,a^\dagger$9

and the generalized anyonic interpolation is

$[a,a^\dagger]=1$0

The two-photon correlation function

$[a,a^\dagger]=1$1

therefore directly records exchange symmetry through bunching or antibunching signatures (Sansoni et al., 2011).

A notable experimental realization encoded exchange symmetry into polarization-entangled Bell states rather than using intrinsically different particles. Bosonic statistics were simulated with

$[a,a^\dagger]=1$2

fermionic statistics with

$[a,a^\dagger]=1$3

and anyonic statistics with

$[a,a^\dagger]=1$4

The corresponding integrated walk network implemented a discrete-time quantum walk with Hadamard coin

$[a,a^\dagger]=1$5

step unitary $[a,a^\dagger]=1$6, a 3D directional-coupler geometry with $[a,a^\dagger]=1$7, $[a,a^\dagger]=1$8, $[a,a^\dagger]=1$9, chip length $\hat{n}=a^\dagger a$0, and single-photon similarity $\hat{n}=a^\dagger a$1; the measured two-photon similarities were $\hat{n}=a^\dagger a$2 and $\hat{n}=a^\dagger a$3 (Sansoni et al., 2011).

A common misconception is that bosonic interference in such circuits requires interactions. The passive-network description shows the opposite: the defining structure is coherent superposition of indistinguishable paths under exchange-symmetrized amplitudes. In this sense, bosonic circuit behavior is already fully visible at the level of linear mode transformations and Fock-space symmetrization.

3. Qumodes, Gaussian circuits, and native bosonic gate sets

In continuous-variable architectures, bosonic quantum circuits operate on qumodes rather than qubits. Gaussian gates are generated by Hamiltonians at most quadratic in quadratures and are compactly described by affine symplectic transformations

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

which update the mean vector and covariance matrix as $\hat{n}=a^\dagger a$5 and $\hat{n}=a^\dagger a$6 (Dutta et al., 2024). The standard single-mode Gaussian gates are displacement

$\hat{n}=a^\dagger a$7

phase rotation $\hat{n}=a^\dagger a$8, and squeezing

$\hat{n}=a^\dagger a$9

while two-mode Gaussian gates include the beam splitter and two-mode squeezing (Dutta et al., 2024).

Gaussian control is central but not universal. The same literature states that non-Gaussian elements such as Kerr evolution, cubic phase, or mode-selective phase operations provide universality and are needed to simulate anharmonic chemistry or realize arbitrary bosonic control (Dutta et al., 2024). In circuit QED, this principle appears in two native universal gate sets. One is based on echoed conditional displacement and qubit rotations,

$$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$0

with

$$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$1

and the other is based on SNAP gates

$$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$2

combined with oscillator displacements (Dutta et al., 2024).

These gate sets support bosonic variational simulation. For electronic structure, the qubit Hamiltonian obtained after Jordan–Wigner mapping is compiled either as a linear combination of echoed conditional displacement blocks or directly into stacked SNAP-displacement circuits, with truncation $$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$3 for an $$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$4-qubit operator embedding (Dutta et al., 2024). The same paper reports that for $$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$5 the ECD route uses $$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$6 terms of depth $$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$7 per grouped operator, whereas the SNAP route compiles each Pauli word with depth $$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$8 and requires 14 Hadamard tests rather than 120 (Dutta et al., 2024).

Bosonic circuit representation is also native to trapped-ion processors. There, a universal feature-embedding circuit uses carrier and red-sideband pulses

$$a=\frac{x+ip}{\sqrt{2}},\qquad a^\dagger=\frac{x-ip}{\sqrt{2}},$$9

and overlap estimation uses a constant-depth circuit with exactly two controlled beam splitters,

$[x,p]=i$0

with readout formula $[x,p]=i$1 (Nguyen et al., 2021).

A software abstraction of this gate model appears in Bosonic Qiskit, which represents each qumode by a register of $[x,p]=i$2 qubits truncating the Fock ladder at $[x,p]=i$3, and implements gates such as $[x,p]=i$4, $[x,p]=i$5, $[x,p]=i$6, $[x,p]=i$7, $[x,p]=i$8, and $[x,p]=i$9 within a hybrid qubit–qumode circuit interface (Stavenger et al., 2022).

4. Encoded bosons on qubit hardware

A distinct branch of the subject represents bosonic dynamics on qubit devices. One route digitizes bosonic fields in a discretized field-amplitude basis. For each lattice site, a field window $R=(x_1,p_1,\ldots,x_N,p_N)^T$0 is sampled at $R=(x_1,p_1,\ldots,x_N,p_N)^T$1 grid points, encoded in $R=(x_1,p_1,\ldots,x_N,p_N)^T$2 qubits, with spacings

$R=(x_1,p_1,\ldots,x_N,p_N)^T$3

The discrete field and conjugate-field operators are related by finite Fourier transform, and the representation error on the low-energy subspace obeys

$R=(x_1,p_1,\ldots,x_N,p_N)^T$4

while the earlier DVR analysis gives

$R=(x_1,p_1,\ldots,x_N,p_N)^T$5

Both formulations state exponential accuracy once the grid exceeds the low-energy cutoff, and both emphasize that field-amplitude digitization makes diagonal-in-$R=(x_1,p_1,\ldots,x_N,p_N)^T$6 interactions polynomially compilable through QFTs and phase rotations (Macridin et al., 2021, Macridin et al., 2018).

Another route encodes bosonic lattice models directly in compact qubit registers. The Binary Encoded Multilevel Particles Ansatz uses $R=(x_1,p_1,\ldots,x_N,p_N)^T$7 qubits per site to encode local occupations $R=(x_1,p_1,\ldots,x_N,p_N)^T$8 and preserves particle count by construction through symmetry-preserving two- and three-qubit gates. The qubit-form number operator is

$R=(x_1,p_1,\ldots,x_N,p_N)^T$9

and the two primitive generators are

$U$0

which commute with $U$1 by design (Bahrami et al., 2024).

A third route is algebraic rather than variational. In the hard-core boson formalism, a qubit is represented by local operators

$U$2

with on-site relations $U$3, $U$4, but inter-site commutation $U$5 for $U$6. Pauli operators become

$U$7

and standard gates acquire direct bosonic expressions, such as

$U$8

This preserves tensor-product structure without sign corrections and yields a matrix-free circuit simulator whose GHZ benchmarks are reported to be at least three orders of magnitude faster than Qiskit’s state-vector simulator in the figures presented (Emmanuel-Costa et al., 26 Jun 2026).

A related but conceptually different construction simulates passive bosonic linear optics through fermion-to-qubit encoding. There, $U$9 bosons in $U$0 modes are represented on $U$1 qubits by an antisymmetrically entangled fermionic state across internal labels; bundle occupations recover the bosonic output distribution, and the Hong–Ou–Mandel coincidence probability under partial distinguishability is

$U$2

with $U$3 encoded by circuit parameters (Chin et al., 2022).

These encoded approaches clarify that bosonic circuit representation is not synonymous with oscillator hardware. It can also mean a resource-efficient qubit description that preserves bosonic commutation, occupation constraints, or linear-optical amplitudes.

5. Matrix functions, Gaussian bosonic complexity, and sampling hardness

A central computational theme is that bosonic circuits often realize amplitudes governed by combinatorial matrix functions. Passive single-photon linear optics yields permanents of interferometer submatrices; Gaussian boson sampling yields hafnians or loop-hafnians depending on squeezing and displacement; and commuting Ising-type spin circuits can be arranged so that leading-order transition amplitudes are proportional to the same functions (Kang et al., 2 Feb 2026). The permanent is

$U$4

the hafnian sums over perfect matchings of a symmetric $U$5 matrix, and the loop-hafnian generalizes hafnian by allowing diagonal self-loops. The special relation

$U$6

makes the permanent a bipartite restriction of the hafnian, while $U$7 when $U$8 has zero diagonal (Kang et al., 2 Feb 2026).

A common misconception is that bosonic circuit complexity is exhausted by permanents. The hafnian and loop-hafnian constructions show a broader hierarchy. In the commuting-XX Ising representation,

$U$9

for the hafnian, permanent, and loop-hafnian cases respectively, with exact circuit implementation because all XX terms commute and the gate count scales as $B[U]$0 (Kang et al., 2 Feb 2026).

Gaussian bosonic circuits admit a different compression. A framework for $B[U]$1 modes encodes the mean vector and covariance matrix of a Gaussian state into an $B[U]$2-qubit system, where the extra qubit labels position versus momentum sectors. Passive Gaussian gates become unitary “real-time” evolutions on the qubit register, while active particle-nonpreserving Gaussian gates become “imaginary-time” evolutions implemented by LCU and post-selection (Barthe et al., 2024). For passive circuits on exponentially many modes, the same work defines a BQP-complete Gaussian-bosonic decision problem and reports numerical simulation of an interferometer on approximately $B[U]$3 billion modes using a $B[U]$4-qubit circuit (Barthe et al., 2024).

The complexity perspective therefore cuts across hardware distinctions. Whether the bosonic circuit is realized in photonics, in a spin model, or in an exponentially compressed qubit encoding, the same matrix functions and hardness mechanisms recur.

6. Superselection, bosonic codes, and fault-tolerant architectures

Particle-number superselection rules provide a different circuit semantics. In the SSRC formulation, physically allowed unitaries satisfy $B[U]$5, and pure states are confined to fixed-$B[U]$6 sectors. Any SSRC pure state

$B[U]$7

can be “extracted” into $B[U]$8 dual-rail qubits encoded in $B[U]$9 orthogonal modes by projecting onto the one-photon-per-external-mode sector. Gaussian, number-preserving operations then map to local qubit gates after extraction, whereas Kerr-like interactions such as

$a,a^\dagger$00

become entangling qubit gates; for appropriate local phase corrections, the induced nonlocal phase is $a,a^\dagger$01, so $a,a^\dagger$02 yields a controlled-$a,a^\dagger$03 (Descamps et al., 2024).

In circuit QED, bosonic quantum circuit representation is also the foundation of bosonic quantum error correction. The basic hardware model is a high-$a,a^\dagger$04 cavity mode dispersively coupled to a nonlinear ancilla with effective Hamiltonian

$a,a^\dagger$05

which enables conditional phases, parity mapping, Wigner tomography, SNAP gates, and ancilla-mediated control (Joshi et al., 2020). Prominent encodings include cat codes, binomial codes, and GKP codes. For example, the four-component cat code uses

$a,a^\dagger$06

with logical $a,a^\dagger$07, while parity measurement is performed by a Ramsey sequence at interaction time $a,a^\dagger$08 (Joshi et al., 2020). The review reports break-even QEC for four-component cat codes and describes code-independent entangling operations such as eSWAP and bilinear beam-splitter couplings (Joshi et al., 2020).

A recent two-mode extension builds a bosonic qubit from the inverse quantum Fourier transform over the real Pauli group $a,a^\dagger$09. For $a,a^\dagger$10, the two-mode Fourier cat codewords factorize as products of single-mode cat states,

$a,a^\dagger$11

with logical operators $a,a^\dagger$12, $a,a^\dagger$13, phase gate

$a,a^\dagger$14

and controlled-$a,a^\dagger$15

$a,a^\dagger$16

The same construction uses beamsplitter-based code deformation to realize $a,a^\dagger$17 and gives an experimentally friendly universal gate set (Leverrier, 22 May 2025).

Current directions extend bosonic circuit representation beyond digital logic to analog simulation. In one circuit-QED proposal, the spin-$a,a^\dagger$18 Heisenberg model is mapped via a polynomially truncated Holstein–Primakoff transformation into an extended Bose–Hubbard Hamiltonian realized by a Josephson-junction array, with parameter identification $a,a^\dagger$19 and $a,a^\dagger$20 (Dudinets et al., 4 Jul 2025). In another superconducting platform, the bosonic Kitaev chain is implemented in synthetic dimensions by mapping lattice sites to cavity frequency modes and generating complex hopping and pairing by parametric pumps at mode-difference and mode-sum frequencies, producing chiral transport, quadrature wavefunction localization, and strong sensitivity to boundary conditions (Busnaina et al., 2023).

Taken together, these developments show that bosonic quantum circuit representation is simultaneously a language for interference, a gate model for oscillator hardware, a bridge from bosons to qubits, and a framework for error-corrected logical computation. Gaussian structure, exchange symmetry, occupation constraints, and non-Gaussian resources remain the recurring themes that determine which bosonic circuits are simulable, which are universal, and which expose classically intractable amplitudes.