Methodology for replacing indirect measurements with direct measurements

In quantum computing, the indirect measurement of unitary operators such as the Hadamard test plays a significant role in many algorithms. However, in certain cases, the indirect measurement can be reduced to the direct measurement, where a quantum state is destructively measured. Here, we investigate under what conditions such a replacement is possible and develop a general methodology for trading an indirect measurement with sequential direct measurements. The results can be applied to construct quantum circuits to evaluate the analytical gradient, metric tensor, Hessian, and even higher order derivatives of a parametrized quantum state. Also, we propose a method to measure the out-of-time-order correlator based on the presented protocol. Our protocols can significantly reduce the depth of a quantum circuit by making the controlled operation unnecessary, and thus are suitable for quantum-classical hybrid algorithms on near-term quantum computers.

Figure 1

Simplest Hadamard test. In the figure, $b\in \{0,1\}$ and $U$, $H$, $S$ are an arbitrary quantum gate, the Hadamard gate, and $e^{-i\pi Z/4}$, respectively. When $b=0$, $\langle Z\rangle =\mathrm{Re}\langle {\psi }_{\text{in}}\left|U\right|{\psi }_{\text{in}}\rangle$ and when $b=1$, $\langle Z\rangle =\mathrm{Im}\langle {\psi }_{\text{in}}\left|U\right|{\psi }_{\text{in}}\rangle$.

Figure 2

(a) Hadamard test with two controlled gates. In the figure, $W$ is an arbitrary quantum gate. (b), (c) Quantum circuits to estimate the output of (a) with direct measurement. ${\mathcal{M}}_G$ is the projective measurement of $G$.

Figure 3

Quantum circuit for the estimation of $\frac{\partial \langle A\left(\mathit{\theta }\right)\rangle }{\partial {\theta }_j}$. (a) The circuit with an ancillary qubit from [15]. $Z_{\mathrm{anc}}$ is the Pauli $Z$ acting only on the ancillary qubit. The output of the circuit, ${\langle Z_{\text{anc}}A\rangle }_j$, is related to the gradient by $\langle Z_{\mathrm{anc}}A\rangle =-\frac{\partial \langle A\left(\mathit{\theta }\right)\rangle }{\partial {\theta }_j}$. (b) Quantum circuit which has equivalent output as (a); ${\langle Z_{\text{anc}}A\rangle }_j={\langle Z_{\text{anc}}\rangle }_j$.

Figure 4

Quantum circuit for the estimation of the real and imaginary parts of the metric tensor $g_{jk}$. (a) Indirect method from Refs. [18, 19]. $b\in \{0,1\}$. When $b=0$, ${\langle Z_{\mathrm{anc}}\rangle }_{jk,0}=4\mathrm{Re}\left(g_{jk}\right)$ and when $b=1$, ${\langle Z_{\mathrm{anc}}\rangle }_{jk,1}=4\mathrm{Im}\left(g_{jk}\right)$. (b) Direct method to estimate the real part of $g_{jk}$ [see Eq. (9)]. (c) Direct method to estimate the imaginary part of $g_{jk}$ [see Eq. (10)].

Figure 5

Indirect approach to measure the OTOC of operators $A$ and $B$ from Ref. [28]. In the figure, $U\left(t\right)=e^{-iHt}$.

Figure 6

Hadamard test with three controlled gates (a) and its replacement (b)–(e). In the figure, $b\in \{0,1\}$ and $U$, $H$, $S$ are an arbitrary quantum gate, the Hadamard gate, and $e^{-i\pi Z/4}$, respectively. When $b=0$, $\langle Z\rangle =\mathrm{Re}\langle {\psi }_{\text{in}}\left|U\right|{\psi }_{\text{in}}\rangle$ and when $b=1$, $\langle Z\rangle =\mathrm{Im}\langle {\psi }_{\text{in}}\left|U\right|{\psi }_{\text{in}}\rangle$.