Cross-Platform verification of Intermediate Scale Quantum Devices: Difference between revisions

Cross-Platform verification of Intermediate Scale Quantum Devices (edit)

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 * <math>U_1, U_2</math>: Quantum operation prepared in <math>S_1</math> and <math>S_2</math> respectively
 * <math>\rho_1, \rho_2</math>: Density matrices of <math>U_1</math> and <math>U_2</math> respectively
+* <math>\rho_{i,A_i}</math>: Reduced density matrices
+* <math>A_i</math>: Sub system of identical size <math>N_{A}</math> where <math>A_i \subseteq S_i</math>
+* <math>N_{A}</math>: Size of subset <math>A_i</math>. Subsets <math>A_1</math> and <math>A_2</math> have the equal size <math>N_{A}</math>
+* <math>D_A</math>: Associated Hilbert space dimension, <math>D_A = d^{N_A}</math>
+* <math>U_A</math>: Random unitary
+* <math>U_k</math>: Local random unitaries acting on spins <math>k=1, .., N_A</math>. Here, the <math>U_k</math> are sampled independently from a [[unitary 2-design]] defined on the local Hilbert space <math>C^d</math>
+* <math>k</math>: Spin
+* <math>|s_A \rangle</math>: This denotes a string of possible measurement outcomes for spins. <math>|s_A \rangle =|s_1, ..., S_{N_A} \rangle </math>
+* <math>P_U^{(i)}</math>: Estimate of probabilities of measurement outcome for different spins.
+* <math>\overline{P_U^{(i)}(s_A)}</math>: The ensemble average over random unitaries of the form <math>U_A</math>
+* <math>D[s_A, s_A']</math>:  Hamming distance between two strings <math>s_A, s_A'</math> is defined as the number of spins where <math>s_k \neq s_k'</math> i.e. <math>D[s_A,s_A'] = |\{k \in \{ 1, .., N_A\}| s_k \neq s_k'\}|</math>
+* <math>F_{max}(\rho_1, \rho_2)</math>: Cross platform fidelity between two quantum states
 ==Hardware Requirements==
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 * The present protocol scales, although exponentially, much more favorably with the (sub)system size, allowing practical cross-platform verification for (sub)systems involving tens of qubits on state-of-the-art quantum devices.
 * This protocol can be used to perform fidelity estimation towards known target theoretical states, as an experiment-theory comparison.
-* In practice, from a finite number of projective measurements performed per random unitary (<math>N_M</math>), a statistical error of the estimated fidelity arises. With that, a finite number (<math>N_U</math>) of random unitaries used to infer overlap and purities can also cause a  statistical error while estimating fidelity. Therefore, the scaling of the total number of experimental runs $N_MN_U$, which are required to reduce this statistical error below a fixed value of <math>\epsilon</math>
+* In practice, from a finite number of projective measurements performed per random unitary (<math>N_M</math>), a statistical error of the estimated fidelity arises. With that, a finite number (<math>N_U</math>) of random unitaries used to infer overlap and purities can also cause a  statistical error while estimating fidelity. Therefore, the scaling of the total number of experimental runs <math>N_MN_U</math>, which are required to reduce this statistical error below a fixed value of <math>\epsilon</math>
 ==Protocol Description==
+'''Input''': <math>S_1, S_2</math> with spins <math>N_1, N_2</math>
+'''Output''': Cross platform fidelity <math>F_{max}</math>
+* Prepare <math>U_1, U_2</math> with density matrices <math>\rho_1, \rho_2</math> in <math>S_1</math> and <math>S_2</math> respectively.
+* Denote <math>\rho_{i,A_i}</math> in <math>D_A</math> as <math>tr_{S_i / A_i} (\rho_i)</math> for <math>A_i \subseteq S_i (i = 1,2)</math>
+* For <math>1, ...., N_u</math>
+** For spin <math>k= 1, ..., N_A</math>:
+*** Sample <math>U_k</math> independently from a [[unitary 2-design]]
+*** Send <math>U_k</math> classically to <math>S_1</math> and <math>S_2</math>
+** For <math>i= 1,2</math>:
+*** Apply <math>U_A = \bigotimes_{k=1}^{N_A}U_k</math> to <math>\rho_{i,A_i}</math> in <math>S_i</math>
+*** For <math>1, ..., N_M</math>:
+**** Perform projective measurements in a computational basis in <math>S_i</math>
+**** Obtain <math>|s_A \rangle</math>
+*** Get estimates of probabilities <math>P_U^{(i)}(s_A) = Tr_{A_i}[U_A \rho_{i,A_i}U^{\dagger}_A |s_A\rangle\langle s_A|]</math>
+* Obtain <math>\overline{P_U^{(i)}(s_A)}</math> and <math>\overline{P_U^{(j)}(s_A)}</math>
+* Define Tr<math>[\rho_{i, A_i}, \rho_{j, A_j}] = d^{N_A}\sum_{s_A,s_A'} (-d)^{-D[s_A, s_A']}\overline{P_U^{(i)}(s_A)P_U^{(j)}(s_A')}</math>
+* For density matrix overlap:
+** Substitute <math>i = 1, j = 2</math>, to get Tr<math>[\rho_{1, A_1}, \rho_{2, A_2}]</math>
+* For the purities of the quantum states:
+** Substitute <math>i =j = 1</math> for Tr<math>\rho_{1, A_1}^2</math>
+** Substitute <math>i =j = 2</math> for Tr<math>\rho_{2, A_2}^2</math>
+* Calculate <math>F_{max}</math> using: <math>F_{max} = \frac{Tr[\rho_{1, A_1}, \rho_{2, A_2}]}{max\{Tr\rho_{1, A_1}^2, Tr\rho_{2, A_2}^2\}}</math>
 ==Further Information==
+* In principle, the cross-platform fidelity can be determined from full quantum state tomography of the two quantum devices. However due to the exponential scaling with the (sub)system size, this approach is limited to only a few degrees of freedom, In contrast, as demonstrated below, the present protocol scales, although exponentially, much more favorably with the (sub)system size, allowing practical cross-platform verification for (sub)systems involving tens of qubits on state-of-the-art quantum devices.
 ==Related Papers==