Scattering Processes from Quantum Simulation Algorithms for Scalar Field Theories

Abstract

We provide practical simulation methods for scalar field theories on a quantum computer that yield improved asymptotics as well as concrete gate estimates for the simulation and physical qubit estimates using the surface code. We achieve these improvements through two optimizations. First, we consider a finite volume approach for estimating the elements of the S-matrix. This approach is appropriate in general for 1+1D and for certain low-energy elastic collisions in higher dimensions. Second, we implement our approach using a series of different fault-tolerant simulation algorithms for Hamiltonians formulated both in the field occupation basis and field amplitude basis. Our algorithms are based on either second-order Trotterization or qubitization. The cost of Trotterization in occupation basis scales as $O(λN^7 |Ω|^3/(M^{5/2} ε^{3/2}))$ where $λ$ is the coupling strength, $N$ is the occupation cutoff, $|Ω|$ is the volume of the spatial lattice, $M$ is the mass of the particles and $ε$ is the uncertainty in the energy calculation used for the $S$-matrix determination. Qubitization in the field basis scales as $O(|Ω|^2 (k^2 Λ+kM^2)/ε)$ where $k$ is the cutoff in the field and $Λ$ is a scaled coupling constant. We find in both cases that the bounds suggest physically meaningful simulations can be performed using on the order of $4\times 10^6$ physical qubits and $10^{12}$ $T$-gates which corresponds to roughly one day on a superconducting quantum computer with surface code and a cycle time of 100 ns. This places the simulation of scalar field theory within striking distance of the gate counts for the best available chemistry simulation results.

Figures (10)

• Figure 1: a) The low lying spectrum of the $\varphi^4$ in its broken phase ($m^2=-0.25$, $\lambda=0.15$). The ground state energy has been subtracted. From Fig. 10 of Bajnok2016. b) The scattering phase inferred from the energies of the two kink states presented in Fig. \ref{['EnergyLevels']}. From Fig. 11 of Bajnok2016.
• Figure 2: a) The $T$ gate count on a $\log$ axis as a function of the field amplitude cutoff $k$ . Algorithm IIIb is plotted with a dotted line as the precise scaling rests upon conjectured behavior of the field operator binary decomposition (Conjecture \ref{['conj:phi24']})). Unknown constant prefactors for the conjecture only for Algorithm IIIb have been set to 1 here. b) The $T$ gate count on a $\log$ axis. as a function of the field occupation cutoff $N$. For both, we consider a strong-coupling regime $\lambda =M = 1$, with $|\Omega| = 10^2$ and $\epsilon_E = 10^{-2}$. .
• Figure 3: The $T$ gate count on a $\log$ axis as a function of the momentum volume cutoff $|\Omega|$. Here we consider a strong-coupling regime $\lambda =M = 1$ and $\epsilon_E = 10^{-2}$. Here we have $k = 20$. By dimensional analysis we expect an approximate scaling relation of $N \sim \sqrt{k}$. In order to illustrate the proper scaling relations (and optimum nature of the amplitude basis) with respect to $|\Omega|$, we select a larger (and therefore more accurate to the physics) value for $N = 9$. Algorithm IIIb is again plotted with a dotted line as the precise scaling rests upon conjectured behavior (Conjecture \ref{['conj:phi24']}).
• Figure 4: a) The logical qubit on a $\log$ axis as a function of the field amplitude cutoff $k$ . Algorithm IIIb is plotted with a dotted line as the precise scaling rests upon conjectured behavior of the field operator binary decomposition (Conjecture \ref{['conj:phi24']})). Unknown constant prefactors for the conjecture have been set to 1 here. b) The exact logical qubiT-gate count on a $\log$ axis. as a function of the field occupation cutoff $N$. For both, we consider a strong-coupling regime $\lambda =M = 1$, with $|\Omega| = 10^2$ and $\epsilon_E = 10^{-2}$.
• Figure 5: The logical qubit count on a $\log$ axis as a function of the momentum volume cutoff $|\Omega|$. Here we consider a strong-coupling regime $\lambda =M = 1$ and $\epsilon_E = 10^{-2}$. Here we have $k = 20$. By dimensional analysis we expect an approximate scaling relation of $N \sim \sqrt{k}$. In order to illustrate the proper scaling relations (and optimum nature of the amplitude basis) with respect to $|\Omega|$, we select a larger (and therefore more accurate to the physics) value for $N = 9$.

Theorems & Definitions (54)

• Claim 1
• Theorem 2
• Proposition 3
• Theorem 4
• Theorem 5
• Lemma 6
• Lemma 7
• Lemma 8: Complexity of noninteracting Hamiltonian Simulation
• proof
• Lemma 9: Eigenbasis for $\boldsymbol{\mathcal{G}_1}$ and $\boldsymbol{\mathcal{G}_2}$