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First and second quantized digital quantum simulations of bosonic systems

Mathias Mikkelsen, Hubert Okadome Valencia

TL;DR

This work addresses the resource trade-offs in simulating bosonic systems by comparing first quantized (U1Q, B1Q) and second quantized (U2Q, B2Q) mappings for systems with $N$ particles in $M$ modes, focusing on $k$-RDM off-diagonal terms and two canonical Hamiltonians. It develops analytic expressions for Pauli-string counts and gate costs, supplemented by numerical analyses on the Bose-Hubbard model and a harmonic oscillator with short-range interactions. The results show that first quantized mappings are generally more gate-efficient than their second quantized counterparts, with unary first quantized mapping typically offering the best gate-efficiency and binary first quantized mapping achieving favorable qubit counts when $M$ is large, though HO scenarios can still be challenging under trotterization. These findings guide mapping choices for bosonic simulations on NISQ and early FTQC devices, highlighting the practicality of first quantized bosonic simulations under particle-number conservation and the continued relevance of second quantization when particle number is not conserved.

Abstract

We compare the basic resource requirements for first and second quantized bosonic mappings in a system consisting of $N$ particles in $M$ modes. In addition to the standard binary first quantized mapping, we investigate the unary first quantized mapping, which we show to be the most gate-efficient mapping for bosons in the general case, although less qubit-efficient than binary mappings. Our comparison focuses on the $k$-body reduced density matrix ($k$-RDM) as well as two standard bosonic Hamiltonians. The first quantized mappings use less resources for off-diagonal terms of the $k$-RDM by a factor of $ \sim N^k$, compared to the second quantized mappings. The number of gates for the first quantized binary mapping increases faster with $M$ compared to the other mappings. Nevertheless, a detailed numeric analysis reveals that the binary first quantized mapping still requires fewer gates than the binary and unary second quantized ones for realistic combinations of $N$ and $M$, while requiring exponentially fewer qubits than the unary mappings. Additionally, the number of CNOT and $R_z(φ)$ gates necessary to express the exponential of the Hamiltonian in the binary first quantized mapping is comparable to the (overall most efficient) unary first quantized one when $M = 2^n$ for both the Bose-Hubbard model and the harmonic trap with short-range interactions. This suggests that this mapping can be both qubit- and gate-efficient for practical problems.

First and second quantized digital quantum simulations of bosonic systems

TL;DR

This work addresses the resource trade-offs in simulating bosonic systems by comparing first quantized (U1Q, B1Q) and second quantized (U2Q, B2Q) mappings for systems with particles in modes, focusing on -RDM off-diagonal terms and two canonical Hamiltonians. It develops analytic expressions for Pauli-string counts and gate costs, supplemented by numerical analyses on the Bose-Hubbard model and a harmonic oscillator with short-range interactions. The results show that first quantized mappings are generally more gate-efficient than their second quantized counterparts, with unary first quantized mapping typically offering the best gate-efficiency and binary first quantized mapping achieving favorable qubit counts when is large, though HO scenarios can still be challenging under trotterization. These findings guide mapping choices for bosonic simulations on NISQ and early FTQC devices, highlighting the practicality of first quantized bosonic simulations under particle-number conservation and the continued relevance of second quantization when particle number is not conserved.

Abstract

We compare the basic resource requirements for first and second quantized bosonic mappings in a system consisting of particles in modes. In addition to the standard binary first quantized mapping, we investigate the unary first quantized mapping, which we show to be the most gate-efficient mapping for bosons in the general case, although less qubit-efficient than binary mappings. Our comparison focuses on the -body reduced density matrix (-RDM) as well as two standard bosonic Hamiltonians. The first quantized mappings use less resources for off-diagonal terms of the -RDM by a factor of , compared to the second quantized mappings. The number of gates for the first quantized binary mapping increases faster with compared to the other mappings. Nevertheless, a detailed numeric analysis reveals that the binary first quantized mapping still requires fewer gates than the binary and unary second quantized ones for realistic combinations of and , while requiring exponentially fewer qubits than the unary mappings. Additionally, the number of CNOT and gates necessary to express the exponential of the Hamiltonian in the binary first quantized mapping is comparable to the (overall most efficient) unary first quantized one when for both the Bose-Hubbard model and the harmonic trap with short-range interactions. This suggests that this mapping can be both qubit- and gate-efficient for practical problems.

Paper Structure

This paper contains 10 sections, 25 equations, 5 figures, 1 table.

Table of Contents

  1. introduction
  2. Mappings, reduced density matrices and basic hamiltonians
  3. Generic Hilbert space mapping
  4. Second quantized mapping
  5. First quantized mappings
  6. Resource comparisons
  7. $k$-RDM and density interactions
  8. Bose Hubbard model and harmonic oscillator
  9. Discussion and outlook
  10. Comparing with optimized CNOT gate counts

Figures (5)

  • Figure 1: Resource comparisons for symmetric $1$-RDM ODTs $\hat{a}^\dagger_k\hat{a}_{l}+\hat{a}^\dagger_{l}\hat{a}_k$. Aside from the B1Q mapping all calculations are for $\hat{a}^\dagger_0\hat{a}_1+\hat{a}^\dagger_1\hat{a}_0$. For the B1Q mapping a calculation for this element corresponding to the minimum Hamming distance and one at indices corresponding to the maximum Hamming distance are both shown. (a)-(c) corresponds to the number of CNOT gates required to express one Trotter step of it's exponential, with (a) and (b) as a function of $M$ for $N=3$ and $N=7$ respectively, while (c) shows the number of CNOT gates normalized to the number required for the U1Q mapping as a function of $N$ for $M=32$. (d)-(f) shows the number of $R_z(\phi)$ gates required for the exponential for the same physical parameters. (g)-(i) shows the number BWCP groups for the same parameters.
  • Figure 2: The same plots as in Fig.\ref{['fig:1RDMresourcecomparison']}, but for symmetric $2$-RDM ODTs $\hat{a}^\dagger_k \hat{a}^\dagger_l \hat{a}_m \hat{a}_n+\hat{a}^\dagger_n \hat{a}^\dagger_m \hat{a}_l \hat{a}_k$. Aside from the B1Q mapping all calculations are for $\hat{a}^\dagger_0 \hat{a}^\dagger_2 \hat{a}_1 \hat{a}_3+\hat{a}^\dagger_3 \hat{a}^\dagger_1 \hat{a}_2 \hat{a}_0$. For the B1Q mapping a calculation for this element corresponding to the minimum Hamming distance and one at indices corresponding to the maximum Hamming distance are both shown.
  • Figure 3: Resource comparisons for the BHM as a function of $M$. (a),(b),(c) corresponds to $N=3$ while (d),(e),(f) corresponds to $N=16$. (a,d) corresponds to the number of CNOT gates required to express one Trotter step of the exponential of the Hamiltonian, while (b,e) shows the corresponding number of $R_z(\phi)$ rotation gates, and (c,f) shows the number of BWCP groups
  • Figure 4: Resource comparisons for the HO as a function of $M$. (a),(b),(c) corresponds to $N=3$ while (d),(e),(f) corresponds to $N=6$. (a,d) corresponds to the number of CNOT gates required to express one Trotter step of the exponential of the Hamiltonian, while (b,e) shows the corresponding number of $R_z(\phi)$ rotation gates, and (c,f) shows the number of BWCP groups
  • Figure 5: Comparing the number of CNOT gates required to express the exponential for (a) the $1$-RDM ODTs, (b) the $2$-RDM ODTs and (c) the BHM as a function of $M$. In addition to the raw CNOT gate counts the CNOT gate count after running the Qiskit optmizer is displayed by the unfilled symbols.