Practical Quantum Circuit Implementation for Simulating Coupled Classical Oscillators

TL;DR

This work presents a concrete quantum-circuit framework for simulating the dynamics of 1D chains of classical coupled oscillators by mapping the Newtonian dynamics to a Hamiltonian evolution and implementing $e^{-i\mathcal{H}t}$ via block-encoding and QSVT. The approach combines block-encoding of the coupling matrix $B$, LCU composition to form a block-encoding of the Hamiltonian $\mathcal{H}$, and QSVT-based polynomial approximations to realize time evolution, with ROAA boosting success probability. Gate-count analyses show favorable scaling for uniform systems ($\mathcal{O}(\log^2 N)$) and more demanding but still polynomial scaling for heterogeneous systems ($\mathcal{O}(N\log N)$), using $\mathcal{O}(\log N)$ qubits. Numerical simulations validate the method by reproducing velocity trajectories and, with post-processing, displacements that agree with classical solvers, highlighting potential for larger-scale quantum studies on future hardware, pending error-mitigation and fault-tolerance developments.

Abstract

Simulating large-scale coupled-oscillator systems presents substantial computational challenges for classical algorithms, particularly when pursuing first-principles analyses in the thermodynamic limit. Motivated by the quantum algorithm framework proposed by Babbush et al., we present and implement a detailed quantum circuit construction for simulating one-dimensional spring-mass systems. Our approach incorporates key quantum subroutines, including block encoding, quantum singular value transformation (QSVT), and amplitude amplification, to realize the unitary time-evolution operator associated with simulating classical oscillators dynamics. In the uniform spring-mass setting, our circuit construction requires a gate complexity of $\mathcal{O}\bigl(\log_2^2 N\,\log_2(1/\varepsilon)\bigr)$, where $N$ is the number of oscillators and $\varepsilon$ is the target accuracy of the approximation. For more general, heterogeneous spring-mass systems, the total gate complexity is $\mathcal{O}\bigl(N\log_2 N\,\log_2(1/\varepsilon)\bigr)$. Both settings require $\mathcal{O}(\log_2 N)$ qubits. Numerical simulations agree with classical solvers across all tested configurations, indicating that this circuit-based Hamiltonian simulation approach can substantially reduce computational costs and potentially enable larger-scale many-body studies on future quantum hardware.

Figures (17)

• Figure 1: A high-level workflow illustrating our circuit-based quantum algorithm for simulating a one-dimensional chain of coupled oscillators. (1) Define the classical system, specifying the initial condition $\left(\vec{x}(0),\dot{\vec{x}}(0)\right)$, masses $\{m_i\}_{i=1}^N$ and spring constants $\{k_{(i,j)}\}_{i,j=1}^{N}$, whose connectivity structure can be encoded by the incidence matrix $\Phi_F$. (2) Following Babbush_2023, rewrite the classical Newtonian equations of motion in a matrix form and then reformulate them as a Hamiltonian simulation problem with associated Hamiltonian $\mathcal{H}$, using mass‐weighted state variable $\vec{y}(t)=\sqrt{M}\,\vec{x}(t)$ (Section \ref{['sec:CCO']}). (3) Build a block encoding circuit of $\mathcal{H}$ (via block encoding circuit of $\sqrt{M}^{-1}$, $\Phi_F$, $\sqrt{W}$) (Section \ref{['subsec:BE_B']} and \ref{['subsec:BE_H']}). (4) Apply QSVT and linear‐combination‐of‐unitaries (LCU) techniques to implement a block encoding circuit of $e^{-i\mathcal{H}t}$ (Section \ref{['subsec:BE_eiht']}). (5) Robust oblivious amplitude amplification (ROAA) boosts the success probability of measuring the correct time-evolved state. (Section \ref{['subsec:fullHS']}). (6) Finally, we extract the time‐evolved state vector, read off velocities directly, and (optionally) reconstruct displacements via a finite‐difference procedure. Classical and quantum results are then compared in post‐processing (Section \ref{['sec:results_exp']}).
• Figure 2: A $(\alpha,a,\varepsilon)$-block-encoding of an $s$-qubit operator $A$. The top $a$ qubits are ancillary (ancilla register), initially in state $\ket{0^a}$. The bottom $s$ qubits (signal register) hold the state on which $A$ acts. Measuring the ancillas in the state $\ket{0^a}$ projects onto $A/\alpha$.
• Figure 3: Circuit for the projector-controlled phase gate $\Pi(\phi)$. An $e^{-\,i\phi Z}$ rotation is flanked by two projector-controlled NOT gates $\mathrm{C}_{\Pi}\mathrm{NOT}$. Multi-controlled gates arise naturally if $\Pi$ projects onto more than one qubit.
• Figure 4: A schematic quantum eigenvalue transformation circuit of $s$-qubit operator $A$ with its $(\alpha_A, a, 0)$--block-encoding $U_A$. By interleaving $U_A$ and $U_A^\dagger$ with projector-controlled phase gates, one effectively applies a polynomial $P(\lambda)$ to the eigenvalues $\lambda$ of $A$. Measuring the ancillas in $\ket{0^a}$ then projects onto the subspace corresponding to $P(A)$. In standard QSVT, non-Hermitian operators require singular-value transformations, but for Hermitian $A$, we can speak directly of eigenvalue transformations (QET). Note that $H$ here represents the Hadamard gate.
• Figure 5: (a) Circuit realizations of a reflection operator $R_0(m)$ and the Grover iteration operator $W$. The reflection operator $R_0(m)$ applies a phase of $-1$ only if the input is $\ket{0^x}$. It decomposes into $X$ and multi-controlled $Z$ gates. (b) The Grover iteration operator $W$ from Definition \ref{['def:AA_W_iteration']} uses two reflections, $R_{\psi_0}$ and $R_{\psi_g}$, along with the block-encoding $U_A$ and its inverse $U_A^\dagger$. Iterating $W$ amplifies the amplitude on $\ket{0^a}\otimes \tilde{A}\ket{v}$.

Theorems & Definitions (15)

• Definition 1: Block Encoding
• Definition 2: Projector-controlled phase gate
• Definition 3: C$_{\Pi}$NOT gate
• Theorem 4: Quantum Singular Value Transformation
• proof : Sketch of Proof
• Definition 5: Grover Iteration
• Theorem 6
• Corollary 7
• Proposition 8
• proof : Proof