Fig. 3: Calculating state overlaps.
From: Simulating groundstate and dynamical quantum phase transitions on a superconducting quantum computer
a The overlap of two translationally invariant states parametrised by U and \(U^{\prime}\) is given by \(\mathop{\lim }\limits_{n\to \infty }{C}_{n}\to {\lambda }^{n}\), where λ is the principle eigenvalue of the transfer matrix delimited by the red dotted line. Cn is evaluated on circuit by measuring the probability of \({\left|0\right\rangle }^{\otimes (n+1)}\) at the output. In order to correct for depolarisation errors, we divide by the Loschmidt echo obtained by evaluating the circuit at \(U^{\prime}=U\). b Overlaps Cn(UA, UB) and Loschmidt echo Cn(UA, UA) evaluated on Rainbow as a function of the order of power method n. c The ratio Cn+1/Cn obtained from the data in (b). The overlap begins to converge as the circuit depth—measured by n—rises. By n = 4 the measured value overlaps within error bars with the exact value of the principal eigenvalue of the transfer matrix (aside from the outlier at n = 6 which occurs due to an error in the estimate of the Loschmidt echo. This is corrected by the interpolation shown in (b). The circuit depth increases with n leading to increased error bars. However, the result is still within errors suggesting that useful information can be extracted even from these deeper circuits. d A demonstration that stochastic optimisation of UB using SPSA converges to UB = UA.
