Hamming Weight-Preserving Quantum Circuits

• Hamming weight-preserving quantum circuits are designed to conserve the total Hamming weight, confining quantum evolution to subspaces with a fixed number of qubit '1's.
• They enable efficient state preparation and amplitude encoding in fixed-weight subspaces, which is critical for simulating symmetry-constrained systems and quantum machine learning.
• These circuits reduce complexity and improve trainability by restricting the parameter space, thereby mitigating barren plateau effects in variational quantum algorithms.

A Hamming weight-preserving quantum circuit is a quantum circuit whose unitaries exactly conserve the total Hamming weight of computational-basis states, confining quantum evolution to subspaces of fixed population (number of qubit "1"s). These circuits are also called "subspace preserving," "energy-conserving" (in contexts where Z-eigenvalues are interpreted as energies), or "particle-number-conserving" in fermionic simulation. Hamming weight-preserving gates and variational ansätze exhibit distinctive expressivity, circuit complexity, and trainability properties. Recent developments reveal their utility in quantum machine learning, data encoding, and efficient simulation of symmetry-constrained physical systems, notably via architectures such as subspace preserving quantum convolutional neural networks and symmetry-aware VQE (Monbroussou et al., 27 Sep 2024, Yan et al., 6 Dec 2024, Monbroussou et al., 2023).

1. Definition and Structural Properties

Let $\mathcal{H} = (\mathbb{C}^2)^{\otimes n}$ denote the $n$-qubit Hilbert space. The Hamming-weight operator is defined as

$W = \sum_{i=1}^n |1\rangle\langle 1|_i,$

and the computational basis $\{|x\rangle:\;x\in\{0,1\}^n\}$ decomposes into subspaces of fixed Hamming weight: $$\mathcal{H} = \bigoplus_{k=0}^n \mathcal{H}_k,\qquad \mathcal{H}_k = \mathrm{span}\{\;|x\rangle\::\:HW(x)=k\;\}.$$ Projectors onto these subspaces satisfy $P_kP_{k'} = \delta_{k,k'}P_k$ and $\sum_{k=0}^n P_k = \mathbb{I}$.

A unitary $U$ is called Hamming weight-preserving if it commutes with $W$: $$[U,W]=0 \Longleftrightarrow U = \bigoplus_{k=0}^n U^{(k)}, \qquad U^{(k)}: \mathcal{H}_k\to\mathcal{H}_k,$$ i.e., $U$ is block-diagonal and never mixes different weight sectors (Monbroussou et al., 27 Sep 2024, Bai et al., 2023).

2. Gate Sets and Circuit Construction

The canonical elementary gate is the two-qubit Reconfigurable Beam Splitter (RBS): $$\mathrm{RBS}(\theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & \cos\theta & -\sin\theta & 0 \ 0 & \sin\theta & \cos\theta & 0 \ 0 & 0 & 0 & 1 \end{pmatrix}$$ in the basis $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$. This generator commutes with $Z\otimes I + I\otimes Z$ and thus strictly preserves total weight.

Circuits are typically composed of layered patterns of RBS gates ("pyramid," "butterfly," "X" layouts), enabling the realization of arbitrary real orthogonal transformations within fixed-weight subspaces (Monbroussou et al., 27 Sep 2024, Farias et al., 30 May 2024). More general constructions exploit XX+YY interaction ("XY model") gates (Bai et al., 2023) or balanced-symmetric (BS) gate variants for complex-valued transforms (Yan et al., 6 Dec 2024). Every such gate acts nontrivially only on the $\{|01\rangle, |10\rangle\}$ subspace, fixing $|00\rangle, |11\rangle$.

The expressivity of these circuits is quantified by the dynamical Lie algebra they generate. Full SU($\dim(\mathcal{H}_k)$) can be reached with suitable gate choices and all-to-all connectivity (see (Yan et al., 6 Dec 2024), Theorem 1). Nearest-neighbor universality is also attainable under mild conditions on the gate generators.

3. Data Encoding and State Preparation

Efficient amplitude encoding into $\mathcal{H}_k$ is central to applications in quantum machine learning (Farias et al., 30 May 2024, Li et al., 20 Aug 2025, Monbroussou et al., 2023). Classical data $x\in\mathbb{R}^{d}$ with $d=\binom{n}{k}$ is mapped to

$$|\psi\rangle = \frac{1}{\|x\|_2}\sum_{j=1}^d x_j|b_j\rangle,$$

where $|b_j\rangle$ runs over all weight-$k$ basis states. Farias et al. (Farias et al., 30 May 2024) present an optimal sequential RBS-gate algorithm, generating an exact amplitude encoding using only $d-1$ real parameters (complex case: $2d-1$). This circuit can be compiled to $O(kd)$ CNOTs.

Further, recent work gives log-depth, size-optimal state preparation circuits for all Hamming-weight-preserving states, matching lower bounds on circuit size and depth as a function of $(n,k)$, and allowing trade-offs between ancillary qubits and depth (Li et al., 20 Aug 2025). For $k=2$, such states correspond to graphs: grid/tree-structured cases can be realized ancilla-free, while general graphs require $O(m)$ ancillas ($m=$ edges), but $O(\log n)$ depth.

4. Quantum Machine Learning Architectures

Quantum convolutional and pooling layers can be realized within the Hamming-weight-preserving framework (Monbroussou et al., 27 Sep 2024). Data tensors are encoded as fixed-weight quantum states. Quantum convolutional layers apply local blocks of RBS-based orthogonal circuits, exactly mimicking classical convolution by processing $K^2$-size pixel blocks. Measurement-based pooling combines amplitude from pixel neighborhoods while retaining global Hamming weight symmetry (except in classical postprocessing). Dense layers are realized as orthogonal transformations in these subspaces.

Quantum subspace-preserving neural architectures exhibit polynomial forward-pass circuit-depth savings over classical counterparts. For instance, a $K\times K$ quantum convolutional layer has depth $O(\log K)$ (classical: $O(K^2)$ multiplications per output), and quantum pooling reduces to constant depth, compared to $O(d^2)$ (with $d=$ input size) classically. Parameter count per quantum filter is substantially lower due to the structure of the fixed-weight block (Monbroussou et al., 27 Sep 2024).

Empirical results on MNIST, Fashion-MNIST, and CIFAR-10 show that subspace-preserving QCNNs match or outperform classical baselines with fewer parameters and polynomial circuit-depth advantages.

5. Expressivity, Trainability, and Avoidance of Barren Plateaus

The expressivity of a Hamming-weight-preserving circuit is governed by the controllability of $\mathcal{H}_k$. Orthogonal transformations on $\mathcal{H}_k$ require $O(d_k^2)$ circuit parameters, but practical tasks (e.g., ground state preparation for VQE) typically need only $O(d_k)$ (Yan et al., 6 Dec 2024, Monbroussou et al., 2023). The dynamical Lie algebra analysis confirms when a given two-qubit gate set is universal within a fixed-weight subspace (Yan et al., 6 Dec 2024).

Trainability is enhanced due to the reduced dimensionality of the subspace: the variance of loss-function gradients over random initializations decays only as $O(1/d_k)$, not $O(1/2^n)$ as in generic variational circuits (Monbroussou et al., 2023). This mitigates concentration of measure and precludes the onset of barren plateaus (exponentially vanishing gradients) for circuits with sufficiently small $k$. Full parameter controllability can be certified using the rank of the quantum Fisher information matrix (QFIM), which is almost everywhere maximal in parameter space (Monbroussou et al., 2023).

6. Computational and Physical Relevance

Hamming-weight-preserving circuits provide polynomial or exponential resource savings for machine learning, data compression, and simulation of fermionic/particle-conserving Hamiltonians (Farias et al., 30 May 2024, Yan et al., 6 Dec 2024). In condensed-matter and quantum chemistry, these circuits naturally encode particle-number symmetries and support hardware-efficient variational ansätze for energy minimization in VQE (Bai et al., 2023, Yan et al., 6 Dec 2024).

For circuit synthesis, the resource-optimality is proven both for general amplitude encoding and for the simulation of arbitrary subspace-preserving unitaries. The key scaling is

• Gate count: $O(kd)$ for state preparation in $\mathcal{H}_k$ (Farias et al., 30 May 2024)
• Circuit depth: $O(\log d)$ possible with sufficient ancillas (Li et al., 20 Aug 2025)
• Exact synthesis of arbitrary energy-conserving unitaries: $O(4^n n^{3/2})$ gates, near the information-theoretic minimum (Bai et al., 2023)

Universal subspace-preserving ansätze can achieve chemical accuracy in electronic structure tasks across molecular and Hubbard models, outperforming standard hardware-efficient ansätze (Yan et al., 6 Dec 2024).

7. Algorithmic and Implementation Constraints

The practical realization of Hamming-weight-preserving circuits requires precise gate compilation. Two-qubit RBS or XY-type gates, layered in minimal architectures, enable both universal and application-specialized ansätze. Implementation constraints arise from connectivity (nearest-neighbor vs. all-to-all), hardware gate sets, and ancillary space (needed for simultaneous log-depth and minimal size (Li et al., 20 Aug 2025)). In platform-specific mapping (e.g., ion traps, superconducting qubits with native XY interaction), circuit depth, gate count, and robustness to noise become interdependent (Farias et al., 30 May 2024).

Notable optimization strategies include:

• Minimal ancilla circuits, trading off depth and parallelism (Zi et al., 9 Apr 2024, Li et al., 20 Aug 2025)
• State preparation "fan-out" and unary encoding with log-depth trees of two-qubit gates
• Error mitigation enhancements for NISQ hardware, exploiting the subspace concentration of amplitude (Farias et al., 30 May 2024)
• Efficient classical simulation for fixed small $k$, as circuit complexity grows polynomially in $n$ for amplitude encoding in $\mathcal{H}_k$ (Monbroussou et al., 2023)

Table: Comparison of quantum convolutional vs. classical convolutional layers (Monbroussou et al., 27 Sep 2024)

Layer Type	Classical (asymptotic)	Quantum HWP circuit (asymptotic)
Convolution	$O(C^2 K^2 \sum_j d_j)$	$O(\max_j\log K_j)$
Average-pooling	$O(C \sum_j d_j^2)$	$O(1)$
Dense (param. $p$)	$O(p\sum_j d_j)$	$O(p)$ (lower parameters)

• Subspace Preserving Quantum Convolutional Neural Network Architectures (Monbroussou et al., 27 Sep 2024)
• Universal Hamming Weight Preserving Variational Quantum Ansatz (Yan et al., 6 Dec 2024)
• Quantum encoder for fixed Hamming-weight subspaces (Farias et al., 30 May 2024)
• Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning (Monbroussou et al., 2023)
• Preparation of Hamming-Weight-Preserving Quantum States with Log-Depth Quantum Circuits (Li et al., 20 Aug 2025)
• Synthesis of Energy-Conserving Quantum Circuits with XY interaction (Bai et al., 2023)
• Shallow Quantum Circuit Implementation of Symmetric Functions with Limited Ancillary Qubits (Zi et al., 9 Apr 2024)