Efficient Classical Simulation of the DQC1 Circuit with Zero Discord

TL;DR

This work shows that zero-discord instances of the DQC1 circuit can be classically simulated in polynomial time provided the target unitary is gate-implementable, by connecting the trace of the unitary to diagonal elements of a Hermitian operator $O$ and constructing a tractable Hamiltonian $H$ via BCH. The method hinges on expressing the DQC1 unitary as $O = -\frac{2}{\pi}H + I^{\otimes n}$, where $H$ is a sum of Pauli strings with poly($n$) terms, enabling efficient classical sampling of diagonal entries to estimate the trace. Schmidt-rank arguments (via Lieb-Robinson bounds) show the final state remains poly($n$)-controlled, supporting Vidal-type simulations and underscoring that discord need not be the sole resource behind speedups in mixed-state computation. Overall, the results reinforce quantum discord as a meaningful resource in mixed-state quantum computing while clarifying the boundary where zero-discord cases remain classically tractable.

Abstract

A path for efficient classical simulation of the DQC1 circuit that estimates the trace of an implementable unitary under the zero discord condition [Phys. Rev. Lett. 105, 190502 (2010)] is presented. This result reinforces the status of non-classical correlations quantified by quantum discord and related measures as the key resource enabling exponential speedups in mixed state quantum computation.

Figures (6)

• Figure 1: The DQC1 circuit for evaluating the normalized trace of the unitary $U_n$ acting on the bottom register of $n$-qubits which are initialized in the fully mixed state.
• Figure 2: Probability, $P_q$ of measuring $| 00\cdots00 \rangle$ at the output of a randomly generated 12-qubit zero discord unitary, when $| 00\cdots0 \rangle$ is the input state is plotted against $N_-$. Also shown as the solid line (red) is the analytic expression for the expectation value of $P_q$ from Eq. \ref{['eq:var']}. We see that the numerically obtained points are distributed around the expected value with small variance.
• Figure 3: Probability, $P_q$ of measuring $| 00\cdots00 \rangle$ at the output of a randomly generated $n$-qubit zero discord unitary, with $| 00\cdots0 \rangle$ as the input state, plotted against $N_{-1}$. The solid line (red) is the analytical expression for the expectation value of $P_q$ from Eq. \ref{['eq:var']} while the dots (Blue) are the numerical obtained data points. For the plots on the left, $n=8$ and for those on the right, $n=12$. The unitary is of the form $V^{\dagger} D V$ where $V$ consists of $m$ layers, each constructed using sets of commuting gates randomly from the Clifford + $T$ gate set. The first, second and third rows correspond to $m = 4$, $m = 8$ and $m = 12$ layers respectively. We see that as $m$ increases the convergence of the values to the expected parabolic curve improves. Further, the convergence is better for larger $N$ for a given $m$.
• Figure 4: The $L_2$ norm of the difference between the exact unitary and the unitary constructed from a Hamiltonian obtained using the BCH formula is plotted. For the plots on the left-hand side the expansion of the unitary was in terms of the gate set (CNOT + H + T) and on the right-hand side it was expanded using (Toffoli + H). The first, second and third rows shows convergence of the $L_2$ norm under BCH expansion for varying numbers of qubits, different number of layers in the circuit implementation of the unitary and the order of BCH truncation respectively. The sample size is 100 for each plot and the order of the BCH truncation for the first two rows is 60.
• Figure 5: Normalized trace for randomly generated hermitian unitary circuits for $n$ = 15 and 18 with $m$ = 10 and 8 respectively is computed using the classical simulation algorithm and plotted (red dots). For each data point, 500 randomly selected diagonal elements of the unitary are average over. The data points have substantial overlap since many of the randomly generated unitaries have the same trace. The dotted line denotes the actual values of the traces and we see that the algorithm performs very well.