Unary First Quantized Mapping

• Unary First Quantized Mapping is an encoding method that represents each boson with an M-qubit register using a one-hot scheme, ensuring proper symmetry of the bosonic state.
• It maps bosonic creation and annihilation operators onto qubit operators, providing gate-efficient implementations for k-body reduced density matrices and standard model Hamiltonians.
• The approach trades off qubit efficiency for significant reductions in gate complexity, making it ideal for digital quantum simulations in fixed-particle systems.

Unary first quantized mapping (U1Q) is an approach for encoding the quantum state of $N$ indistinguishable bosons in $M$ orthogonal single-particle modes onto qubit registers, particularly suited for digital quantum simulations in fixed-particle-number systems. In the U1Q scheme, each boson is represented by a separate “mode–register” consisting of $M$ qubits, employing a one-hot encoding for mode occupation, and ensuring evolution and preparation only within the totally symmetric Hilbert space. This mapping is distinguished by its substantial gate efficiency, specifically for simulating few-body reduced density matrices and standard bosonic Hamiltonians, while trading off qubit efficiency compared to binary-encoded alternatives.

1. Encoding Scheme and State Representation

The U1Q mapping assigns an $M$-qubit register $\mathcal{H}_\alpha$ to each individual particle $\alpha = 1, ..., N$, yielding a total Hilbert space dimension of $(\mathbb{C}^M)^{\otimes N}$. Within each register, the occupation of mode $l \in \{0, ..., M-1\}$ by boson $\alpha$ is encoded by the basis state: $$|\alpha, l\rangle \longleftrightarrow |0\cdots0\,\underset{l}{1}\,0\cdots0\rangle$$ which is a one-hot representation—precisely one qubit in each register is “1,” the remainder are “0.” Only fully symmetric register states are physically permitted, reflecting bosonic permutation symmetry.

Creation and annihilation operators in this framework are mapped within each register as: $$|\alpha, l\rangle\langle\alpha, m| = \hat S_{\alpha,l}^+\,\hat S_{\alpha,m}^-, \qquad \hat S_{\alpha,j}^\pm = \frac{1}{2}(X_{\alpha, j} \pm i Y_{\alpha, j})$$ Thus, the many-body operator $a_l^\dagger a_m$ acts as a sum over individual-particle transitions.

2. Qubit Counts and Mapping Comparisons

The U1Q mapping requires: $Q_{\rm U1Q} = N \times M$ qubits. Comparative qubit requirements for other mappings are as follows:

Mapping	Qubit count
U1Q	$N\,M$
B1Q	$N\,\lceil \log_2 M \rceil$
U2Q	$M\,(N+1)$
B2Q	$M\,\lceil\log_2(N+1)\rceil$

This quantifies the trade-off: U1Q achieves minimal gate complexity but at the cost of a linear scaling in both $N$ and $M$ for qubit resources, contrasting with logarithmic scaling in binary schemes.

3. Gate Counts for $k$-Body Reduced Density Matrices

The number of gates required to implement $k$-body off-diagonal reduced density matrix (RDM) exponentials in U1Q is determined by the quantity and length of generated Pauli strings: $$\#\text{Pauli strings} = 2^{2k}N^k, \qquad \text{string length} = 2k$$ Each such string is implemented using a CNOT-staircase circuit of cost $2(2k-1)$ CNOTs. Consequently: $$n_{R_z}^{(k\text{-RDM})} = 2^{2k}N^k, \qquad n_{\rm CNOT}^{(k\text{-RDM})} = 2(2k-1)\,2^{2k}N^k$$ For the unary second quantized (U2Q) mapping, corresponding gate counts exponentially increase: $$n_{R_z}^{(k)}|_{\rm U2Q} = (4N)^{2k}, \qquad n_{\rm CNOT}^{(k)}|_{\rm U2Q} = 2(4k-1)(4N)^{2k}$$ U1Q yields a savings factor $\sim (2N)^{2k}$ in both rotation and CNOT counts over U2Q for $k$-body RDMs. Asymptotic bounds for binary mappings are less favorable, though goodness-of-fit depends on $M$; for practical $N \leq 10, M \leq 32$, numeric results indicate that U1Q beats B1Q by factors of $2$–$10$ in gate counts.

4. Resource Costs for Model Hamiltonians

Bose–Hubbard Model

For the Bose–Hubbard Hamiltonian on $M$ sites,

$$H_{\rm BH} = -J\sum_{j=0}^{M-1}(a_j^\dagger a_{j+1} + \text{H.c.}) + \frac{U}{2}\sum_{j=0}^{M-1} n_j(n_j-1)$$

U1Q implementation requires: $$n_{R_z}^{\rm BH}|_{\rm U1Q} = 2M N + \frac{1}{2}M N(N-1)$$

$n_{\rm CNOT}^{\rm BH}|_{\rm U1Q} = 4M N + M N(N-1)$

Compared to U2Q, which scales as $8M N^2$ $R_z$ and $48M N^2$ CNOTs, this reveals savings of order $N$ to $N^2$ in gates.

Harmonic Trap Model

For the harmonic oscillator plus contact interactions,

$$H_{\rm HO} = \sum_{kl}h_{kl}a_k^\dagger a_l + \tfrac{1}{2}\sum_{klmn} V_{klmn} a_k^\dagger a_l^\dagger a_m a_n$$

U1Q deploys: $$n_{R_z}^{\rm HO}|_{\rm U1Q} = O(M^2 N^2), \qquad n_{\rm CNOT}^{\rm HO}|_{\rm U1Q} = O(M^2 N^2)$$ Resource benchmarking in the primary study confirms one to two orders of magnitude fewer $R_z$ and CNOT gates in U1Q than U2Q/B2Q for $N\sim3$–$6,$ $M \lesssim 32$.

5. Comparative Scaling and Efficiency

A summary comparison of mappings:

Mapping	Qubits	$\#$Pauli strings for $k$-RDM ODT	string length	$n_{R_z}$	$n_{\rm CNOT}$
U1Q	$N\,M$	$2^{2k}\,N^k$	$2k$	$2^{2k}N^k$	$2(2k-1)\,2^{2k}N^k$
U2Q	$M(N+1)$	$(4N)^{2k}$	$4k$	$(4N)^{2k}$	$2(4k-1)(4N)^{2k}$
B1Q (upper)	$N\lceil\log_2M\rceil$	$N^k\,2^{k\lceil\log_2M\rceil}$	$k\lceil\log_2M\rceil$	same as strings	$2(k\lceil\log_2M\rceil-1)\times$strings
B2Q (upper)	$M\lceil\log_2(N+1)\rceil$	$N^{2k}2^{2k\lceil\log_2(N+1)\rceil}$	$2k\lceil\log_2(N+1)\rceil$	same as strings	$2(2k\lceil\log_2(N+1)\rceil-1)\times$strings

The mapping is most gate-efficient for off-diagonal $k$-RDMs across all tested particle and mode ranges. U1Q also produces shorter Pauli strings, optimizing circuit depth. For conventional model Hamiltonians, gate counts are reduced by factors of $10$–$100$ against U2Q/B2Q mappings for physically realistic choices of $N$ and $M$.

6. Operator Decomposition and Circuit Implementation

In U1Q, operator exponentials are constructed using Pauli gadgets:

• One-body term: For hopping $a_l^\dagger a_m + \text{H.c.}$, decomposition is

$$\sum_{\alpha=1}^N (X_{\alpha,l} X_{\alpha,m} + Y_{\alpha,l} Y_{\alpha,m})$$

Each exponential of $X_lX_m + Y_lY_m$ requires: - CNOT${}_{\alpha,l\to\alpha,m}$ - $R_z(2\theta)$ on $\alpha,m$ - CNOT${}_{\alpha,l\to\alpha,m}$

• Two-body term: $a_k^\dagger a_l^\dagger a_m a_n + \text{H.c.}$ generates weight-4 Pauli strings, implemented with a CNOT-staircase of depth $6$ plus one $R_z$.
• On-site interaction: $n_j(n_j-1)$ decomposes as $(N+1)$ single-qubit $Z$ terms per site, requiring only $R_z$ rotations and no CNOTs.

Benchmarking figures detail explicit circuit layouts and resource analysis. A plausible implication is that U1Q is particularly attractive for both Noisy Intermediate-Scale Quantum (NISQ) devices and early Fault-Tolerant Quantum Computation (FTQC) scenarios when particle-number conservation is present.

7. Significance and Scope of Application

U1Q is fundamentally constrained by its linear qubit scaling; for systems where qubit availability is a limiting factor, binary schemes remain more attractive. However, for quantum simulators where gate cost predominates or where modest $N$ and $M$ are sufficient, U1Q represents the leading choice for bosonic Hamiltonian simulation in the first quantized paradigm. This suggests ongoing relevance as quantum devices transition to larger system sizes, yet with attention to hardware qubit growth and mapping trade-offs. The methodology is applicable to any bosonic system compatible with particle-number conservation and mode orthogonality, particularly in contexts demanding high gate-efficiency relative to qubit count (Mikkelsen et al., 13 Nov 2025).