Editing Cross-Platform verification of Intermediate Scale Quantum Devices

This [https://arxiv.org/abs/1909.01282 protocol] is used to perform cross-platform verification of quantum simulators and quantum computers. This is used to directly measure the overlap and purities of two quantum states prepared in two different physical platforms and thus used to measure the fidelity of two possibly mixed states. This protocol infers the cross-platform fidelity of two quantum states from statistical correlations between the randomized measurements performed on the two different devices.

'''Tags:''' [[:Category: Certification protocol|Certification Protocol]], [[Cross-Platform Fidelity]], [[Building Trust]], [[Two devices]]

==Assumptions==
* There are no prior assumptions on the quantum states.
* The spin values for the two quantum devices are known.

==Outline==
The aim here to perform cross-platform verification by measuring the overlap of quantum states produced with two different experimental setups, potentially realized on very different physical platforms, without any  prior assumptions on the quantum states themselves. This can be used to
whether two quantum devices have prepared the same quantum state. Here, the cross-platform fidelity is inferred from the statistical correlations between randomized measurements performed on the first and second device. 

This protocol to measure the cross-platform fidelity of two quantum states requires only classical communication of random unitaries and measurement outcomes between the two platforms, with the experiments possibly taking place at very different points in time and space.

This protocol consists of the following steps:

* We start with two quantum devices which are based on different physical platforms, each consisting of two different spins. Two quantum operations are prepared in these quantum devices, which are each described by a density matrix.
* We find the reduced density matrices for the sub-systems of identical size for each device using partial trace operator over that sub-system. 
* We apply a same random unitary is applied to the two quantum states. This random unitary is defined as the product of local random unitaries acting on all spins of the subsystem. Here, the local random unitaries are sampled independently from a [[unitary 2-design]] defined on the local Hilbert space and sent via classical communication to both devices.
* Now projective measurements in a computational basis are performed for both the systems.
* Repeating these measurements for the fixed random unitary provides us with the estimates of probability of measurement outcomes for the both the states.
* This entire procedure is then repeated for many different random unitaries.
* Finally we estimate the density matrix from the second order cross-correlations between the two platforms using the ensemble average of probabilities over random unitaries from the above procedure. 
* The purities for the two sub systems are obtained as second-order auto-correlations of the probabilities.

==Notation==
* <math>N_M</math>: Finite  number  of  projective  measurements  performed per  random  unitary
* <math>N_U</math>: Finite  number of random unitaries used to infer overlap
* <math>\epsilon</math>: Fixed value of statistical error
* <math>S_1, S_2</math>: Two devices realised on different physical platforms
* <math>N_1, N_2</math>: Spins consisted in <math>S_1</math> and <math>S_2</math> respectively
* <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==
* Two quantum devices on two different physical platforms
* Trusted Measurement device.
* Classical communication channel

==Properties==
* '''Figure of merit''': Cross-platform fidelity of two quantum states
* We can estimate the density matrix overlap of two quantum states here as well as their purities.
* 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 <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==
* A.Elben et al (2020) arXiv:1909.01282: Cross-Platform Verification of Intermediate Scale Quantum Devices

<div style='text-align: right;'>''*contributed by Rhea Parekh''</div>