Optimal quantum measurements of expectation values of observables

Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state $\mid \psi ⟩$ of the quantum system. Such expectation values can be measured by repeatedly preparing $\mid \psi ⟩$ and coupling the system to an apparatus. For this method, the precision of the measured value scales as $\frac{1}{\sqrt{N}}$ for $N$ repetitions of the experiment. For the problem of estimating the parameter $\varphi$ in an evolution $e^{-i\varphi H}$, it is possible to achieve precision $\frac{1}{N}$ [the quantum metrology limit; see Giovannetti et al., Phys. Rev. Lett. 96, 010401 (2006)] provided that sufficient information about $H$ and its spectrum is available. We consider the more general problem of estimating expectations of operators $A$ with minimal prior knowledge of $A$. We give explicit algorithms that approach precision $\frac{1}{N}$ given a bound on the eigenvalues of $A$ or on their tail distribution. These algorithms are particularly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of $\mid \psi ⟩$ and $U\mid \psi ⟩$, where $U$ is an implementable unitary operator. We explicitly consider the issue of confidence levels in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.

Figure 1

Quantum network for the one-ancilla algorithm to measure $⟨U⟩=\mathrm{tr}\left(U\rho \right)$ with ${\mid +⟩}_{\mathsf{a}}=({\mid 0⟩}_{\mathsf{a}}+{\mid 1⟩}_{\mathsf{a}})∕\sqrt{2}$ in the logical basis. The desired expectation is given by $\mathrm{tr}\left(U\rho \right)=⟨2{\sigma }_{+}^{\left(\mathsf{a}\right)}⟩=⟨{\sigma }_x^{\left(\mathsf{a}\right)}⟩+i⟨{\sigma }_y^{\left(\mathsf{a}\right)}⟩$, where $⟨{\sigma }_x^{\left(\mathsf{a}\right)}⟩$ and $⟨{\sigma }_y^{\left(\mathsf{a}\right)}⟩$ are the expectations of the Pauli matrices ${\sigma }_x^{\left(\mathsf{a}\right)}$ and ${\sigma }_y^{\left(\mathsf{a}\right)}$ for the final state, which are estimated by repeating the experiment and measuring either ${\sigma }_x^{\left(\mathsf{a}\right)}$ or ${\sigma }_y^{\left(\mathsf{a}\right)}$ on the control (ancilla) qubit labelled $\mathsf{a}$. Because these measurements have $\pm 1$ as possible outcomes, their statistics are determined by the binomial distribution.

Figure 2

(Color online) Geometrical construction for computing $\mathrm{Re}\left(⟨\psi \mid U\mid \psi ⟩\right)$ from $a=\mid ⟨\psi \mid U\mid \psi ⟩\mid$ and $2b_0=\mid (1+⟨\psi \mid U\mid \psi ⟩)\mid$. According to the law of cosines, ${\left(2b_0\right)}^2=a^2+1+2a\cos \left(\theta \right)$, and we have $\mathrm{Re}\left(⟨\psi \mid U\mid \psi ⟩\right)=a\cos \left(\theta \right)=[{\left(2b_0\right)}^2-a^2-1]∕2$.

Figure 3

(Color online) OEA flowchart. An estimate of the overlap $⟨\psi \mid U\mid \psi ⟩$ is obtained. The algorithm requires three state preparations and calls the AEA three times. The amplitude of the returned value shown in the flowchart may need to be adjusted according to the value of $a$ to optimize the precision. For details see the text.

Figure 4

(Color online) Visualization of the parameterization of the overlap in terms of points on the upper hemisphere of a unit sphere. The function $h$ is defined by $h(x_1,x_2,x_3)=x_1+i{x}_2$. Note that for overlaps $\mid ⟨\psi \mid U\mid \psi ⟩\mid$ approaching 1 and small $\delta$, ${\delta }^{\prime }$ approaches ${\delta }^2∕2⪡\delta$.

Figure 5

(Color online) Bloch-sphere representation of the rotations induced on the subspace spanned by $\mid \psi ⟩$ and $U\mid \psi ⟩$ by the operators $S_0$ and $S_1$.

Figure 6

Step (3) of the PEA to estimate bit $k$ of the eigenphase, where $k=3$. The phase ${\widehat{\phi }}_k$ is computed according to previously obtained information about the eigenphase. By applying it before the measurement, the probability of obtaining the optimal value for bit $k$ is maximized. The measurement is denoted by the triangle pointing left with +/− inside and is a measurement in the ∣+⟩/∣−⟩ basis. The outlined part of the network is parallelized in Sec. 5.

Figure 7

Parallelization of the PEA algorithm to estimate the bit $k=3$ of the phase. This replaces the outlined parts of the network in Fig. 6. $E$ is an entangler such that $E{\mid 0⟩}_{\mathsf{a}}^{\otimes 2^{k-1}}=({\mid 0\dots 0⟩}_{\mathsf{a}}+{\mid 1\dots 1⟩}_{\mathsf{a}})∕\sqrt{2}$ and $E{\mid 100\dots 0⟩}_{\mathsf{a}}=({\mid 0\dots 0⟩}_{\mathsf{a}}-{\mid 1\dots 1⟩}_{\mathsf{a}})∕\sqrt{2}$. $E^{-1}$ is the decoding operation that maps $E^{-1}{\mid 0\dots 0⟩}_{\mathsf{a}}={\mid +0\dots 0⟩}_{\mathsf{a}}$, and $E^{-1}{\mid 1\dots 1⟩}_{\mathsf{a}}={\mid -0\dots 0⟩}_{\mathsf{a}}$, where $\mid \pm ⟩=(\mid 0⟩\pm \mid 1⟩)∕\sqrt{2}$. The $k\text{th}$ bit is estimated from the measurement outcome of the first ancilla qubit in the logical basis.