Measuring energy by measuring any other observable

We present a method to estimate the probabilities of outcomes of a quantum observable, its mean value, and higher moments by measuring any other observable. This method is general and can be applied to any quantum system. In the case of estimating the mean energy of an isolated system, the estimate can be further improved by measuring the other observable at different times. Intuitively, this method uses interplay and correlations between the measured observable, the estimated observable, and the state of the system. We provide two bounds: one that is looser but analytically computable and one that is tighter but requires solving a nonconvex optimization problem. The method can be used to estimate expectation values and related quantities such as temperature and work in setups where performing measurements in a highly entangled basis is difficult, finding use in state-of-the-art quantum simulators. As a demonstration, we show that in Heisenberg and Ising models of ten sites in the localized phase, performing two-qubit measurements excludes 97.5% and 96.7% of the possible range of energies, respectively, when estimating the ground-state energy.

Figure 1

Bounds on energy probabilities and the mean energy. Each energy probability $p_{E_l}$ (purple lines) is bounded by $a_l^{max}$ from below (blue bar) and by $b_l^{min}$ (red bar) from above [see Eq. (3)]. The analytical bound on the mean energy $E_{min}^{\mathrm{lin}}\le \langle E\rangle \le E_{max}^{\mathrm{lin}}$ [Eq. (8)] is computed as follows. Imagine a bottle of probabilities with volume 1. To obtain (a) the lower bound on the mean energy, we pour the probabilities on the graph from the bottle and fill the minimal probability of each energy given by $a_l^{max}$ (blue solid bar). We will have some probabilities left in the bottle, so we pour the rest and top the red bars up to their maximum $b_l^{min}$ (red striped bar), starting from the lowest to the highest energy, until we run out of probabilities in the bottle, i.e., until all the probabilities on the graph sum up to 1. Taking the mean value of such a distribution will yield the lower bound [see Eq. (11)]. (b) The upper bound is obtained by the same method but going from the highest to the lowest energy [see Eq. (13)]. The mean energy $\langle E\rangle$ lies somewhere in the shaded region.

Figure 2

Heisenberg model in the localized $\left(W=0.5\right)$ and the delocalized $\left(W=10\right)$ phases, with three particles on six sites. The energy is estimated by measuring in the local number basis (corresponding to $k=0$ in Fig. 3). The initial state is either a ground state (G), a cold state (C), Eq. (19), or a hot state (H). The graphs show the true mean energy $\langle E\rangle$ (single symbols: Star, diamond, and disk for G, C, and H, respectively) and intervals of analytic bounds $[E_{min}^{\mathrm{lin}},E_{max}^{\mathrm{lin}}]$ (solid lines) and numerical bounds $[E_{min},E_{max}]$ (dashed lines) for each state. We also plot the list of energy eigenvalues at the very bottom (E). In the localized phase, energy eigenstates have a large overlap with the local number basis, making the energy estimation significantly more precise. (a) Localized (b) Delocalized.

Figure 3

Heisenberg model in (b) and (c) the localized $\left(W=0.5\right)$ and (d) and (e) the delocalized $\left(W=10\right)$ phases, with five particles on ten sites. The energy is estimated with progressively added optimized $k$-local measurements, with the same initial states as in Fig. 2 (G, C, and H) and energy eigenvalues plotted at the very bottom (E). In large systems, we can no longer employ numerical methods to improve the analytic bound. Instead, we employ different measurements. (a) Sketch of allowed operations. Here $k=0$ (black) corresponds to measuring in the local number basis; $k=1$ (purple) allows for using single-site unitary operators before measuring in the local number basis, which leads to a general single-site measurement; $k=2$ (blue) allows using two-site local operators, etc. (b) and (d) In the ground-state-optimized method, the measurement basis is given by the eigenbasis of the $k$-local reduced state of the ground state [see Eqs. (A1) and (A2)]. Intervals of analytic bounds $[E_{min}^{\mathrm{lin}\left(\mathrm{k}\right)},E_{max}^{\mathrm{lin}\left(\mathrm{k}\right)}]$ are labeled with the colors matching the allowed operations. (c) and (e) In the observable-optimized method, the experimenter measures $k$-local blocks of the Hamiltonian. When $k=10$, the observable-optimized method achieves perfect energy estimation for all initial states $(k=10$ corresponds to measuring the Hamiltonian itself). In contrast, when $k=10$, the ground-state-optimized method achieves that only for the ground state $(k=10$ corresponds to measuring in a basis that contains the ground state). Using two-qubit measurements $\left(k=2\right)$ in the localized phase, the bound excludes 97.5% of the possible range of energies when estimating the ground-state energy. See Appendix pp1 for the descriptions of the ground-state-optimized and observable-optimized methods. See Appendix pp5 for details and specifically Tables 2 and 3 for additional simulations of experimentally achievable XY, PXP, and Ising models.

Figure 4

Illustration of the performance when estimating energy by local Pauli $x$ measurements in a simple qubit example. We show the quality $Q_1$ factor (percentage of excluded energies) when estimating energy given by the Hamiltonian $\widehat{H}={\widehat{\sigma }}_z$ by measuring in the ${\widehat{\sigma }}_x$ basis, for different initial states on the Bloch sphere. (a) Quality factor of the initial state. The value $Q_1=1$ (light orange, which happens for state $|\psi \rangle =|+\rangle )$ corresponds to perfect identification of the energy, while the value $Q_1=0$ (dark blue, which happens for state $|\psi \rangle =|0\rangle )$ corresponds to failure in identifying the energy. (b) Quality factor over initial states if we allow time evolution of the state with the Hamiltonian and measure at different times $t\in [0,\pi /2)$ to build up statistics of outcomes at each time of this interval. The Hamiltonian revolves the state around the $z$ axis, so during the time evolution, the state picks up the best quality factor at that latitude from (a).

Figure 5

Sketch of allowed operations, i.e., $k$-local measurements. Here $k=0$ corresponds to measuring in the local number basis, $k=1$ allows for using single-site unitary operators leading to a general single-site measurement, $k=2$ for two-site local operators, etc.

Figure 6

Observable-optimized type-2 method for the large system of the PXP model. The PXP model is the only one of our considered models in which type 1 and type 2 differ. Note the worse performance for $k=1$ compared to $k=0$ for the cold and hot states and better performance for $k=1,2$ compared to the type-1 method for the ground state.