Average-case Speedup for Product Formulas

Abstract

Quantum simulation is a promising application of future quantum computers. Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. For an accurate product formula approximation, the state-of-the-art gate complexity depends on the number of terms in the Hamiltonian and a local energy estimate. In this work, we give evidence that product formulas, in practice, may work much better than expected. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states, while the existing estimate is for the worst states. For general $k$-local Hamiltonians and higher-order product formulas, we obtain gate count estimates for input states drawn from any orthogonal basis. The gate complexity significantly improves over the worst case for systems with large connectivity. Our typical-case results generalize to Hamiltonians with Fermionic terms, with input states drawn from a fixed-particle number subspace, and with Gaussian coefficients (e.g., the SYK models). Technically, we employ a family of simple but versatile inequalities from non-commutative martingales called $\textit{uniform smoothness}$, which leads to $\textit{Hypercontractivity}$, namely $p$-norm estimates for $k$-local operators. This delivers concentration bounds via Markov's inequality. For optimality, we give analytic and numerical examples that simultaneously match our typical-case estimates and the existing worst-case estimates. Therefore, our improvement is due to asking a qualitatively different question, and our results open doors to the study of quantum algorithms in the average case.

Figures (10)

• Figure 1: Concentration of gate complexity distribution for product formula for states drawn from any fixed orthogonal basis. The vast majority of states are controlled by our typical case results (Theorem \ref{['thm:Trotter_non_random_maintext']}), while extremal states may require the worst case guarantees \ref{['eq:1_norm_main']}thy_trotter_error. The two gate complexities coexist and differ because the Trotter error is a high-dimensional object.
• Figure 2: The local energy estimates $\Vert {\bm{H}} \Vert_{(local),1}$ and $\Vert {\bm{H}} \Vert_{(local),2}$ sum over terms overlapping with a site $i$, and maximize over the sites. This is usually smaller than the global Hamiltonian.
• Figure 3: Trotter error for the all-to-all interacting Heisenberg model for second-order Suzuki formulas $\bm{S}_2(t/r)$. We fix time $t = 10$, repeats $r = 20000$, and change the system size $n = 5,\cdots, 13$. Each Trotter error is estimated by medium-of-mean: take the medium over $27$ bins, where each bin is an average over $32$ independent disorder realization. The fit $a(n+c)^b$ gives the system size dependence $b$. For average inputs (2-norm), the empirical exponent reads $b=2.03\pm 0.03$, which matches the theoretical bound (Theorem \ref{['thm:Trotter_non_random_maintext']}, $b = 2$). For worst inputs (operator norm), the empirical exponent is much larger, $b=4.07\pm 0.13$, which matches the theoretical bound (thy_trotter_error, $b = 4$).
• Figure 4: Time dependence of the 2-norm Trotter error in Figure \ref{['fig:P2_all_XXYYZZ']}. We fix repetition $r = 20000$, the system size $n = 12$, and change time $t = 0, \cdots 17.5$. The fit $a t^b+c$ gives the time dependence exponent $b= 2.74 \pm 0.04$ (variance calculated by independent runs), which deviates slightly from the theoretical upper bounds (Theorem \ref{['thm:Trotter_non_random_maintext']}, $b = 3$).
• Figure 5: Different orders of Suzuki formulas for the all-to-all interacting Heisenberg model. For the first-order Lie-Trotter-Suzuki formula, the parameters are $t = 5, r = 200000, n = 5,\cdots, 14$. We take medium over 8 bins, each averaging over 12 runs. The fit $a(n+c)^b$ gives the empirical system size dependence $b=1.46\pm 0.03$, matching the theoretical bound (Theorem \ref{['thm:Trotter_non_random_maintext']}, $b = 1.5$); the parameters for 4-th order formula are: $t=10, r =1000, n = 5,\cdots, 13$. We take medium over 32 bins, each averaging over 15 runs. The empirical exponent reads $b=2.98\pm 0.03$, matching the theoretical bound (Theorem \ref{['thm:Trotter_non_random_maintext']}, $b = 3$).

Theorems & Definitions (70)

• Theorem 1.1: Trotter error in $k$-local models
• Corollary 1.1.1
• Proposition 1.1.1: A model with different $p$-norms and spectral norm
• Theorem 1.2: (informal) Trotter error in random models
• Proposition 1.2.1: Distinct Hamiltonians
• Proposition 1.4.1: Uniform smoothness for subsystems
• Proposition 2.0.1: Typical states and Schatten p-norms
• proof : Proof of Proposition \ref{['prop:typical_Schatten']}
• Proposition 2.1.1: Uniform smoothness for subsystems
• proof : Proof of Fact \ref{['fact:nc_convexity']}