DQC1-hardness of estimating correlation functions

Abstract

Out-of-Time-Order Correlation function measures transport properties of dynamical systems. They are ubiquitously used to measure quantum mechanical quantities, such as scrambling times, criticality in phase transitions, and detect onset of thermalisation. We characterise the computational complexity of estimating OTOCs over all eigenstates and show it is Complete for the One Clean Qubit model (DQC1). We then generalise our setup to establish DQC1-Completeness of N-time Correlation functions over all eigenstates. Building on previous results, the DQC1-Completeness of OTOCs and N-time Correlation functions then allows us to highlight a dichotomy between query complexity and circuit complexity of estimating correlation functions.

Figures (6)

• Figure 1: DQC1 computation with one clean and $n$ mixed state qubits
• Figure 2: A circuit for $OTOC_{\infty}$ measuring normalised $2k$-point OTOC on $n$-qubit system, $\braket{W_1(t)V_1 \ldots W_k(t)V_k}$ as $Tr[({| 0 \rangle \langle 0 |} \otimes I) C ({| 0 \rangle \langle 0 |} \otimes \frac{I}{2^n}) C^{\dagger} ]$
• Figure 3: Schematic of the $4$-point OTOC circuit described by $\Phi_4[C]$ that estimates normalised trace of circuit $C$.
• Figure 4: Schematic of the $6$-point OTOC circuit described by $\Phi_6[C]$ that estimates normalised trace of circuit $C$.
• Figure 5: Schematic of the $8$-point OTOC circuit described by $\Phi_8[C]$ that estimates normalised trace of circuit $C$.

Theorems & Definitions (10)

• Definition 1: OTOC
• Theorem 1
• Corollary 1
• Definition 2: $N$-time correlation function
• Corollary 2
• Example 1
• Example 2
• Example 3
• Example 4
• Example 5