Quantum computing fidelity susceptibility using automatic differentiation

Automatic differentiation is an invaluable feature of machine learning and quantum machine learning software libraries. In this work, it is shown how quantum automatic differentiation can be used to solve the condensed-matter problem of computing fidelity susceptibility, a quantity whose value may be indicative of a phase transition in a system. Results are presented using simulations including hardware noise for small instances of the transverse-field Ising model, and a number of optimizations that can be applied are highlighted. Error mitigation (zero-noise extrapolation) is applied within the autodifferentiation framework to a number of gradient values required for the computation of fidelity susceptibility and a related quantity, the second derivative of the energy. Such computations are found to be highly sensitive to the additional statistical noise incurred by the error mitigation method.

Figure 1

Graphical depiction of how the quantities of interest depend on quantum gradients. The three gradients in red are obtained through quantum computation using parameter-shift rules, while the derivatives of the variational parameters with respect to $r$ are obtained by solving the response equations. Numbers in parentheses correspond to equations in the main text which are used to obtain the desired quantity.

Figure 2

Circuit for the four-spin variational wave function encoded in two qubits.

Figure 3

Optimized circuit for the six-spin variational wave function encoded in three qubits.

Figure 4

Circuit for calculating the squared overlap of the wave functions instantiated by the unitary operators $U_i$ and $U_f$.

Figure 5

Fidelity susceptibility per site of the transverse-field Ising model as a function of the Hamiltonian parameter $r$ for a lattice of four sites. The circles are for an ideal (noiseless) quantum computation. The squares are results of a noisy simulation. The solid line is the exact result.

Figure 6

Second derivative of the energy per site of the transverse-field Ising model as a function of the Hamiltonian parameter $r$ for a lattice of four sites. The circles are for an ideal (noiseless) quantum computation. The squares are results of a noisy simulation. The solid line is the exact result.

Figure 7

Fidelity susceptibility per site of the transverse-field Ising model as a function of the Hamiltonian parameter $r$ for a lattice of six sites. The circles are for an ideal (noiseless) quantum computation. The squares are results of a noisy simulation. The solid line is the exact result.

Figure 8

Second derivative of energy per site of the transverse-field Ising model as a function of the Hamiltonian parameter $r$ for a lattice of six sites. The circles are for an ideal (noiseless) quantum computation. The squares are results of a noisy simulation. The solid line is the exact result.

Figure 9

Second energy derivative per site for the four-spin system. Results are plotted along the horizontal axis for $r\in [0.5,0.6,...,1.4]$, but offset for readability. The numbers in the legend indicate which scale factors were used in the mitigation process (e.g., “1,3,5” means mitigation was performed using 3 points for the extrapolation: the unmodified circuit, adding one fold, and adding two folds). For each mitigation method, 100 independent trials were run; the error bars correspond to the standard deviation of these results. As expected, a greater number of points in the extrapolation leads to higher variance in the potential result.

Figure 10

Fidelity susceptibility per site for the four-spin system. For each mitigation method, 100 independent trials were run; the error bars correspond to the standard deviation of these results. Greater variation is observed, compared to the second energy derivative of Fig. 9, due to a double dependence on the derivatives of variational parameters with respect to the Hamiltonian parameter $r$.

Figure 11

Second energy derivative per site for the six-spin system. Error bars correspond to the standard deviation of 100 independent trials. Compared to results in the four-spin case, the variance is higher for both types of folding.

Figure 12

Fidelity susceptibility per site for the six-spin system. Error bars correspond to the standard deviation of 100 independent trials. Similar to the four-spin case, the variance in results is higher than that of the second energy derivative. In the case of unitary folding at $r=1.4$, one trial resulted in a markedly large condition number for the matrix inversion involved in the solution of the response equations, leading to significantly less accurate results. We found such behavior to be common for the six-spin case when pushing beyond the use of three scale factors for both types of folding.

Figure 13

Circuit for calculating the overlap of the wave functions instantiated by the unitary operators $U_i$ and $U_f$ using the Hadamard test. The gate $R$ is $R{}_y(-\pi /2)$ or $R{}_x(\pi /2)$ for the real and imaginary parts, respectively.

Figure 14

Fidelity susceptibility of the transverse-field Ising model as a function of the Hamiltonian parameter $r$ for four spins comparing the Hadamard test (squares) with the overlap circuit in Fig. 4 (circles). The solid line is the exact result.

Figure 15

Circuit for calculating the squared overlap of the wave functions instantiated by the unitary operators $U_i$ and $U_f$ using the swap test.

Figure 16

Fidelity susceptibility of the transverse-field Ising model as a function of the Hamiltonian parameter $r$ for four spins using the swap test in Fig. 15 (squares) compared to the overlap circuit in Fig. 4 (circles). Results of using alternate versions of the swap test discussed in [45] are shown using up-triangles (ancilla-based algorithm, Ref. [45], Fig. 5) and down-triangles (Bell-based algorithm, Ref. [45], Fig. 6). The solid line is the exact result.

Figure 17

Error-mitigated ground-state energies of the four-spin system. The mean and standard deviation of 100 independent trials are shown for each value of $r$.

Figure 18

Mitigation error for the ground-state energy of the four-spin system.

Figure 19

Error-mitigated ground-state energies of the six-spin system. The mean and standard deviation of 100 independent trials are shown for each value of $r$.

Figure 20

Mitigation error for the ground-state energy of the six-spin system.

Figure 21

Mitigation error for the second energy derivative of the four-spin system.

Figure 22

Mitigation error for the fidelity susceptibility of the four-spin system.

Figure 23

Mitigation error for the second energy derivative of the six-spin system.

Figure 24

Mitigation error for the fidelity susceptibility of the six-spin system.