Cross-Platform Verification of Intermediate Scale Quantum Devices

We describe a protocol for cross-platform verification of quantum simulators and quantum computers. We show how to measure directly the overlap $\mathrm{Tr}[{\rho }_1{\rho }_2]$ and the purities $\mathrm{Tr}[{\rho }_{1,2}^2]$, and thus a fidelity of two, possibly mixed, quantum states ${\rho }_1$ and ${\rho }_2$ prepared in separate experimental platforms. We require only local measurements in randomized product bases, which are communicated classically. As a proof of principle, we present the measurement of experiment-theory fidelities for entangled 10-qubit quantum states in a trapped ion quantum simulator.

Figure 1

Fidelity estimation with randomized measurements. We present a protocol to measure the fidelity ${\mathcal{F}}_{\max }({\rho }_{1,A_1},{\rho }_{2,A_2})$ of two quantum states described by (reduced) density matrices ${\rho }_{i,A_i}={\mathrm{Tr}}_{{\mathcal{S}}_i\backslash A_i}[{\rho }_i]$ ($i=1$, 2). On two platforms ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$, the quantum states ${\rho }_1$ and ${\rho }_2$ are prepared with quantum operations ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$, respectively. Randomized measurements are performed on both platforms in (sub)systems $A_1\subseteq {\mathcal{S}}_1$ and $A_2\subseteq {\mathcal{S}}_2$ of size $N_A$, implemented with the same local random unitaries $U_1\otimes \dots \otimes U_{N_A}$ which are shared via classical communication (red arrows). From statistical cross-correlations (autocorrelations) of outcome probabilities $P_U^{(i)}({\mathbf{s}}_A)$ ($i=1$, 2), the overlap $\mathrm{Tr}[{\rho }_{1,A_1}{\rho }_{2,A_2}]$ (the purities $\mathrm{Tr}[{\rho }_{i,A_i}^2]$) and thus ${\mathcal{F}}_{\max }({\rho }_{1,A_1},{\rho }_{2,A_2})$ are inferred (see text).

Figure 2

Scaling of the required number of measurements. (a),(b) Average statistical error $|({\mathcal{F}}_{\max }({\rho }_A,{\rho }_A){)}_e-1|$ as a function of the number of measurements $N_M$ per random unitary for various $N_U$ (darkness of colors). The state ${\rho }_A$ of $N_A=8$ qubits ($d=2$) is taken to be (a) a pure product state (PP) and (b) a pure Haar random state (PR). Black lines are guides for the eye, $\sim 1/(N_M\sqrt{N_U})$. (c) Scaling of the minimal number of required measurements $N_M$ to estimate $[{\mathcal{F}}_{\max }({\rho }_A,{\rho }_A){]}_e$ up to a fixed statistical error of 0.05 as a function of the number of qubits $N_A$, for fixed $N_U=100$. The mixed random states (MR) are obtained from tracing out 3 qubits from Haar random states of $N_A+3$ qubits.

Figure 3

Experiment-theory verification in a trapped ion quantum simulator. Measured fidelities ${\mathcal{F}}_{\max }({\rho }_E,{\rho }_T)$ as a function of partition size $N_A$ (total system 10 qubits) for states ${\rho }_E$ evolved with $H_{XY}$ ($J_0=420{\mathrm{s}}^{-1}$, $\alpha =1.24$) for various times; experimental data from Ref. [28]. Theory states ${\rho }_T$ are obtained with (a) unitary dynamics and including (b) decoherence effects (see text). In both panels, $N_U=500$ and $N_M=150$. Error bars are obtained with bootstrap resampling [39]. Dashed lines are guides for the eye.

Figure 4

Experiment self-verification in a trapped ion quantum simulator. (a),(b) Estimated fidelities ${\mathcal{F}}_{\max }$ of two reduced states ${\rho }_{E_1}$ and ${\rho }_{E_2}$ prepared sequentially in the same experiment as a function of partition size, $[1\to N_A]$. The states ${\rho }_{E_1}$ and ${\rho }_{E_2}$ are (a) two Néel states which have been (b) time evolved under $H_{XY}$ ($J_0=420{\mathrm{s}}^{-1}$, $\alpha =1.24$) to $t=1\mathrm{ms}$; experimental data from Ref. [28]. Experiment-theory fidelities are obtained by simulating unitary dynamics (see text). Dashed lines are guides for the eye. (c),(d) Measured fidelities ${\mathcal{F}}_{\max }\mathbf{(}{\rho }_E(t_1),{\rho }_E(t)\mathbf{)}$ for states time evolved with $H_{XY}$ as a function of the time difference $t-t_1$ ($t_1=1\mathrm{ms}$) for (c) a clean system and (d) with additional disorder (see text). Different colors refer to different partitions $[1\to N_A]$. Lines show theory simulations including decoherence effects (see text). In all panels, error bars are estimated with bootstrap resampling [39].