2312.06975v1

# Arbitrary Ground State Observables from Quantum Computed Moments **Authors**: Harish J. Vallury1 and Lloyd C. L. Hollenberg > 0009-0007-3355-5458 0000-0001-7672-6965 School of Physics, The University of Melbourne Parkville, VIC 3010, Australia Email: ## Abstract The determination of ground state properties of quantum systems is a fundamental problem in physics and chemistry, and is considered a key application of quantum computers. A common approach is to prepare a trial ground state on the quantum computer and measure observables such as energy, but this is often limited by hardware constraints that prevent an accurate description of the target ground state. The quantum computed moments (QCM) method has proven to be remarkably useful in estimating the ground state energy of a system by computing Hamiltonian moments with respect to a suboptimal or noisy trial state. In this paper, we extend the QCM method to estimate arbitrary ground state observables of quantum systems. We present preliminary results of using QCM to determine the ground state magnetisation and spin-spin correlations of the Heisenberg model in its various forms. Our findings validate the well-established advantage of QCM over existing methods in handling suboptimal trial states and noise, extend its applicability to the estimation of more general ground state properties, and demonstrate its practical potential for solving a wide range of problems on near-term quantum hardware. ## I Introduction Many difficult problems with practical application in physics and chemistry involve determining the ground state properties of a quantum system. This has been a popular road to follow towards the development of near-term applications of quantum computers, with much of the recent focus primarily on variational quantum algorithms [1, 2, 3]. These hybrid approaches involve the preparation of complicated states designed to represent the ground state of the desired quantum system, via the measurement and minimisation of energy [4, 5, 6]. Most interesting problems at scale require high-depth circuits to prepare such states, and are thus subject to noise which is often severely detrimental to the quality of results [7, 8]. Some time ago, an analytic cluster expansion of the Lanczos recursion was uncovered [9]. Following from this, it was found that for a given quantum system Hamiltonian $H$ , one could compute Hamiltonian moments $\langle H^{k}\rangle$ with respect to a trial state with some overlap with the ground state to produce a powerful approximation to the ground state energy [10, 11], shown below to fourth order in the moments: $$ E_{\rm 0}^{\rm{L}(4)}=c_{1}-\frac{c_{2}^{2}}{c_{3}^{2}-c_{2}c_{4}}\left[\sqrt{ 3c_{3}^{2}-2c_{2}c_{4}}-c_{3}\right], \tag{4} $$ where the cumulants $c_{n}$ are defined as: $$ c_{n}=\langle H^{n}\rangle-\sum_{p=0}^{n-2}\binom{n-1}{p}c_{p+1}\langle H^{n-1 -p}\rangle. \tag{2} $$ The rapid development of quantum computer technology has provided a fitting application for this result, allowing for advanced state preparation and measurement of these Hamiltonian moments in order to solve interesting problems. We refer to this as the quantum computed moments (QCM) method, and recent work applied to problems in quantum magnetism [12, 13] and chemistry [14] has demonstrated the effectiveness of QCM in estimating the ground state energy of a system, even when moments are measured with respect to a suboptimal or noisy trial state. This technique can correct for a low complexity ansatz and a high degree of noise, proving to be remarkably useful on present-day quantum hardware for a variety of different problem instances. In recent times, these QCM studies have been accompanied elsewhere in the literature by a general renewed interest in Hamiltonian moments in a quantum computing context [15, 16, 17, 18], with the primary focus being on the ground state energy problem. However, a problem which is much more practically useful and indeed more computationally complex is that of determining arbitrary ground state observables [19]. It turns out that Hamiltonian moments can also be used to estimate general ground state properties via a Hellman-Feynman approach [20, 21]. Here we develop this technique in the QCM context. When applied to models of interest, our results show that the crucial robustness property of QCM for energy problems indeed carries over to this more general ground state observable procedure. ## II Method Suppose we wish to find the expectation value of some general observable (given by Hermitian operator $A$ ) with respect to the ground state $\ket{\psi_{0}}$ of a Hamiltonian $H$ . Consider the following parameterised Hamiltonian: $$ \displaystyle H_{\lambda}=H+\lambda A \tag{3} $$ The parameterised Hamiltonian $H_{\lambda}$ has ground state $\ket{\psi_{\lambda,0}}$ with energy $E_{\lambda,0}$ which we can approximate with: $$ E_{\lambda,0}\approx E^{{\rm L}(4)}_{\lambda,0}|_{\psi_{t}}. \tag{4} $$ The approximation in Equation (4) is expressed entirely in terms of Hamiltonian moments $\langle\psi_{t}|(H_{\lambda})^{k}|\psi_{t}\rangle$ , up to $k=4$ , with respect to some trial state $\ket{\psi_{t}}$ which has nonzero overlap with $\ket{\psi_{\lambda,0}}$ . Starting with the Hellmann-Feynman theorem, $$ \langle\psi_{\lambda,0}|\,\diff{H_{\lambda}}{\lambda}\,|\psi_{\lambda,0} \rangle=\diff{E_{\lambda,0}}{\lambda}, \tag{5} $$ one can substitute the derivative of the Hamiltonian in Equation (3) and use the approximation in Equation (4) to obtain: $$ \langle\psi_{\lambda,0}|\,A\,|\psi_{\lambda,0}\rangle\approx\frac{\rm d}{ \mathrm{d}\lambda}\left(E^{{\rm L}(4)}_{\lambda,0}|_{\psi_{t}}\right). \tag{4} $$ Thus if we want to compute the observable $A$ with respect to the ground state $\ket{\psi_{0}}$ of $H$ , we can choose $\ket{\psi_{t}}$ to have some overlap with $\ket{\psi_{0}}$ and evaluate the derivative in Equation (6) around $\lambda=0$ : $$ \langle\psi_{0}|\,A\,|\psi_{0}\rangle\approx\frac{\rm d}{\mathrm{d}\lambda} \left(E^{{\rm L}(4)}_{\lambda,0}|_{\psi_{t}}\right)_{\lambda=0}. \tag{4} $$ This leads to a practically useful final expression for the ground state observable: $$ \langle A\rangle_{0}^{\rm L(4)}\equiv\frac{1}{2\varepsilon}\left(E_{+ \varepsilon,0}^{\rm L(4)}-{E_{-\varepsilon,0}^{\rm L(4)}}\right)\approx\langle \psi_{0}|\,A\,|\psi_{0}\rangle, \tag{4} $$ where $\varepsilon>0$ is sufficiently small as to find the $\lambda=0$ derivative, and the $E_{\pm\varepsilon,0}^{\rm L(4)}$ are calculated via Equation (1) from the moments $\langle\,(H\pm\varepsilon A)^{k}\,\rangle$ with respect to some trial state $\ket{\psi_{t}}$ up to $k=4$ . We expand upon the analytic studies of this method, first introduced in [20, 21], by now considering the approach in a quantum computing context. Note that, in practice, the $E_{+\varepsilon,0}^{\rm L(4)}$ and ${E_{-\varepsilon,0}^{\rm L(4)}}$ in Equation 8 are not computed separately, but rather come from the same set of measurements on the quantum computer. The derivative is taken with respect to a parameter only introduced in the classical post-processing stage, so remains unaffected by the stochastic nature of the quantum measurement. We can thus expect to see similar robustness behaviour as has been observed in the QCM energy estimate, and in the following section, we test the effectiveness of this arbitrary observables technique by performing noisy simulations on a variety of trial states and Hamiltonians. ## III Results <details> <summary>x1.png Details</summary>  ### Visual Description ## [Chart/Diagram Type]: Dual-Panel Scientific Plot with Hamiltonian and Correlation Functions ### Overview The image contains two vertically stacked scientific plots, labeled (a) and (b), which analyze the properties of a quantum system on a 2D lattice. Both plots share the same x-axis parameter `x` and are divided into three distinct regions by vertical lines at `x = -0.5` and `x = 0.5`. Each plot compares three different quantities (represented by blue, orange, and green lines) and includes a schematic of a 3x3 square lattice. The overall context appears to be the study of a phase transition or crossover in a quantum many-body system, likely related to the transverse-field Ising model or a similar Hamiltonian. ### Components/Axes **Common Elements:** * **X-axis:** Labeled `x`. Scale ranges from -1.0 to 1.0 with major ticks at -1.0, -0.5, 0.0, 0.5, 1.0. * **Vertical Dividers:** Solid black vertical lines at `x = -0.5` and `x = 0.5`. * **Region Labels (Top of each panel):** Three quantum state notations are placed above the three regions defined by the vertical lines: * Left region (`x < -0.5`): `|ψ₀^{[x=-1]}⟩` * Center region (`-0.5 < x < 0.5`): `|ψ₀^{[x=0]}⟩` * Right region (`x > 0.5`): `|ψ₀^{[x=1]}⟩` * **Lattice Schematic:** A 3x3 grid of circles (sites) connected by lines (bonds) is centered in the lower half of each plot. **Panel (a) Specifics:** * **Title/Label:** `(a)` in the top-left corner. * **Y-axis:** Labeled `Energy`. Scale ranges from -0.55 to -0.35 with major ticks at -0.55, -0.50, -0.45, -0.40, -0.35. * **Legend (Top-Left):** * Blue line: `⟨H⟩` * Orange line: `E₀^{L(4)}` * Green line: `E₀` * **Equation (Top Center):** The Hamiltonian for the system is given as: `H = (1/(4q)) Σ_{<i j>} (Z_i Z_j + x (X_i X_j + Y_i Y_j))` * **Lattice Schematic:** The 3x3 lattice is shown with all sites and bonds in black/white. **Panel (b) Specifics:** * **Title/Label:** `(b)` in the top-left corner. * **Y-axis:** Labeled `Correlation`. Scale ranges from 0.40 to 1.00 with major ticks at 0.40, 0.60, 0.80, 1.00. * **Legend (Top-Left):** * Blue line: `⟨C⟩` * Orange line: `⟨C⟩₀^{L(4)}` * Green line: `⟨C⟩₀` * **Equation (Center):** The correlation function being plotted is defined as: `C_ij = Z_i Z_j` * **Lattice Schematic:** The 3x3 lattice is shown with two specific sites highlighted in red (one at the top-left corner and one at the center of the grid), indicating the pair `(i, j)` for which the correlation `C_ij` is calculated. ### Detailed Analysis **Panel (a) - Energy vs. x:** * **Trend Verification:** * **Blue Line (`⟨H⟩`):** This is a piecewise linear function. It slopes upward from `x = -1.0` to `x = -0.5`, remains constant (flat) at its maximum value from `x = -0.5` to `x = 0.5`, and then slopes downward from `x = 0.5` to `x = 1.0`. * **Orange Line (`E₀^{L(4)}`):** This is a smooth, symmetric, concave-down parabola. It increases from `x = -1.0`, peaks at `x = 0.0`, and then decreases symmetrically to `x = 1.0`. * **Green Line (`E₀`):** This is also a smooth, symmetric, concave-down parabola, very similar in shape to the orange line but consistently slightly lower in energy across the entire range. * **Data Points (Approximate):** * At `x = -1.0`: All three lines converge at approximately `Energy = -0.55`. * At `x = 0.0`: The blue line is at its plateau value of approximately `-0.35`. The orange and green parabolas peak here, with the orange line (`E₀^{L(4)}`) nearly touching the blue line's plateau, and the green line (`E₀`) peaking slightly lower, at approximately `-0.36`. * At `x = 1.0`: All three lines converge again at approximately `Energy = -0.55`. * At the boundaries `x = ±0.5`: The blue line has a sharp corner. The orange and green curves pass through these points smoothly, with values around `-0.40`. **Panel (b) - Correlation vs. x:** * **Trend Verification:** * **Blue Line (`⟨C⟩`):** This is a piecewise constant function with a sharp jump. It is flat at a low value from `x = -1.0` to `x = -0.5`, jumps to a high constant value from `x = -0.5` to `x = 0.5`, and then jumps back down to the low constant value from `x = 0.5` to `x = 1.0`. * **Orange Line (`⟨C⟩₀^{L(4)}`):** This is a smooth, symmetric, concave-down parabola. It increases from `x = -1.0`, peaks at `x = 0.0`, and then decreases symmetrically to `x = 1.0`. * **Green Line (`⟨C⟩₀`):** This is also a smooth, symmetric, concave-down parabola, following the orange line very closely but lying slightly below it across the entire range. * **Data Points (Approximate):** * At `x = -1.0` and `x = 1.0`: The blue line is at its low plateau of approximately `Correlation = 0.30`. The orange and green parabolas start at this same value. * At `x = 0.0`: The blue line is at its high plateau of `Correlation = 1.00`. The orange and green parabolas peak here, with the orange line (`⟨C⟩₀^{L(4)}`) reaching `1.00` and the green line (`⟨C⟩₀`) peaking just below it, at approximately `0.98`. * At the boundaries `x = ±0.5`: The blue line has a vertical jump. The orange and green curves pass through these points smoothly, with values around `0.80`. ### Key Observations 1. **Phase Transition Signatures:** The vertical lines at `x = ±0.5` mark critical points where the behavior of the exact quantities (blue lines) changes abruptly—from sloping to flat in (a), and from low to high in (b). This is characteristic of a first-order phase transition or a sharp crossover. 2. **Approximation Accuracy:** The orange lines (`E₀^{L(4)}` and `⟨C⟩₀^{L(4)}`) appear to be approximations (likely from a 4th-order linked-cluster expansion or similar method) to the exact ground state quantities (green lines: `E₀` and `⟨C⟩₀`). The approximations are excellent, especially near `x=0`, but show slight deviations. 3. **Symmetry:** Both plots are perfectly symmetric about `x = 0.0`, indicating a symmetry in the underlying Hamiltonian (e.g., `x → -x` symmetry). 4. **Correlation Function Behavior:** The correlation `C_ij` for the specific pair highlighted (corner and center sites) is maximal (`1.0`) in the central phase (`|ψ₀^{[x=0]}⟩`) and drops to a constant, lower value (`~0.30`) in the outer phases. This suggests the central phase has strong, possibly long-range, order for this operator. ### Interpretation The data demonstrates the properties of a quantum system governed by the Hamiltonian `H`, which interpolates between different regimes via the parameter `x`. The three regions correspond to distinct quantum phases or ground states, labeled `|ψ₀^{[x=-1]}⟩`, `|ψ₀^{[x=0]}⟩`, and `|ψ₀^{[x=1]}⟩`. * **What the data suggests:** The system undergoes two sharp transitions at `x = ±0.5`. In the outer phases (`|x| > 0.5`), the energy `⟨H⟩` varies linearly with `x`, and the correlation `⟨C⟩` is constant and low. In the central phase (`|x| < 0.5`), the energy is constant (maximized), and the correlation is constant and maximal. This is a classic signature of a system where the ground state changes discontinuously at the critical points. * **Relationship between elements:** The Hamiltonian equation defines the energy being plotted in (a). The correlation function `C_ij = Z_i Z_j` is a specific observable derived from the same Hamiltonian, plotted in (b). The lattice diagrams visually ground the abstract equations, showing the physical system (a 2D square lattice) and, in (b), the specific operator being measured. The state labels at the top identify the ground state wavefunction in each region. * **Notable anomalies/patterns:** The most striking pattern is the "flat-top" behavior of the exact quantities (blue lines) in the central region. This indicates that within this phase, the ground state energy and the specific correlation `⟨C⟩` are *insensitive* to the parameter `x`. This is unusual and points to a highly degenerate or specially symmetric ground state manifold in that phase. The smooth parabolic approximations (orange/green) fail to capture this flatness, highlighting the non-perturbative nature of the transition. The perfect symmetry of the plots confirms the Hamiltonian's symmetry under `x → -x`. </details> Figure 1: (a) Ground state energy estimates for the Heisenberg XXZ model as a function of $x$ in the Hamiltonian. Green shows the exact ground state energy, $E_{0}$ . The energy estimates, $\langle H\rangle$ (blue) and $E_{0}^{\rm L(4)}$ (orange), are evaluated with respect to three trial states: the $x=-1$ (ferromagnetic) ground state for $-1<x<-\frac{1}{2}$ , the $x=0$ (Néel) ground state for $-\frac{1}{2}<x<\frac{1}{2}$ , and the $x=1$ (antiferromagnetic) ground state for $\frac{1}{2}<x<1$ . (b) Estimates of ground state $ZZ$ correlation between two next-nearest-neighbour spins (marked in red) versus $x$ , using the same three trial states. Like before, the exact correlation is shown in green, the direct expectation value in blue, and the moments-based estimate in orange. <details> <summary>x2.png Details</summary>  ### Visual Description ## [Multi-Panel Line Chart]: Expectation Values of Operator M vs. Coupling Parameter g ### Overview The image displays a 3x3 grid of line charts, totaling nine individual plots. Each plot shows the relationship between a coupling parameter `g` (x-axis) and the expectation value of an operator `M` (y-axis) for three different quantum states or calculation methods. The grid is organized by two parameters: `p` (columns) and `F` (rows). The plots demonstrate how the behavior of these expectation values changes as `p` and `F` vary. ### Components/Axes * **Grid Structure:** * **Columns (Parameter `p`):** Labeled at the top of each column as `p = 0.01` (left), `p = 0.10` (center), `p = 0.50` (right). * **Rows (Parameter `F`):** Labeled to the left of each row as `F ≈ 0.99` (top), `F ≈ 0.80` (middle), `F ≈ 0.40` (bottom). * **Individual Plot Axes:** * **X-axis:** Labeled `g` in all plots. The scale runs from `0.00` to `2.00` with major ticks at intervals of `0.25`. * **Y-axis:** Represents the expectation value of operator `M`. The scale runs from `-0.5` to `0.0` with major ticks at intervals of `0.1`. * **Legend (Present in each plot):** * **Blue Line:** `⟨ψ_t^g|M|ψ_t^g⟩` (Expectation value in the time-evolved state `ψ_t^g`). * **Orange Line:** `⟨M⟩_0^{L(4)}|ψ_t^g⟩` (Likely a theoretical or approximate expectation value, possibly from a 4th-order linked-cluster expansion, evaluated for the state `ψ_t^g`). * **Green Line:** `⟨ψ_0^g|M|ψ_0^g⟩` (Expectation value in the initial state `ψ_0^g`). * **Legend Position:** Typically located in the top-right or center-right quadrant of each subplot. ### Detailed Analysis **General Trend Across All Plots:** All three lines (blue, orange, green) start near `0.0` at `g=0.00` and decrease (become more negative) as `g` increases towards `2.00`. The rate of decrease is steepest for small `g` and flattens out for larger `g`. **Analysis by Row (Parameter `F`):** * **Top Row (`F ≈ 0.99`):** The lines are very smooth. The blue line (`⟨ψ_t^g|M|ψ_t^g⟩`) and the orange line (`⟨M⟩_0^{L(4)}|ψ_t^g⟩`) are nearly indistinguishable across all `p` values. The green line (`⟨ψ_0^g|M|ψ_0^g⟩`) is also very close, showing only a slight deviation at higher `g` for `p=0.50`. * **Middle Row (`F ≈ 0.80`):** The blue line begins to show noticeable high-frequency noise or fluctuations, especially for `g > 0.5`. The orange and green lines remain smooth and closely aligned with each other. The separation between the blue line and the orange/green pair becomes more apparent as `p` increases. * **Bottom Row (`F ≈ 0.40`):** The blue line exhibits significant noise and large fluctuations across the entire range of `g`. The orange and green lines remain smooth but start to show a clear separation from each other, particularly at higher `p` values (`p=0.10` and `p=0.50`). For `p=0.50`, the orange line (`⟨M⟩_0^{L(4)}|ψ_t^g⟩`) is consistently above (less negative than) the green line (`⟨ψ_0^g|M|ψ_0^g⟩`). **Analysis by Column (Parameter `p`):** * **Left Column (`p = 0.01`):** The three lines are most tightly clustered. Even at `F ≈ 0.40`, the separation between the orange and green lines is minimal. * **Center Column (`p = 0.10`):** The separation between the blue line and the orange/green pair becomes more pronounced with decreasing `F`. The orange and green lines begin to separate slightly at `F ≈ 0.40`. * **Right Column (`p = 0.50`):** This column shows the most dramatic effects. The blue line is highest (least negative) and most noisy at low `F`. The separation between the orange and green lines is most significant here, especially at `F ≈ 0.40`, where the orange line is clearly above the green line for `g > ~0.25`. ### Key Observations 1. **Noise Correlation with `F`:** The noise in the blue line (`⟨ψ_t^g|M|ψ_t^g⟩`) is inversely correlated with the parameter `F`. High `F` (~0.99) yields smooth curves, while low `F` (~0.40) yields highly fluctuating curves. 2. **Divergence with `p`:** The parameter `p` controls the divergence between the different calculation methods. At low `p` (0.01), all methods agree closely. At high `p` (0.50), significant differences emerge, particularly between the time-evolved state expectation (blue) and the other two, and between the theoretical approximation (orange) and the initial state expectation (green) at low `F`. 3. **Agreement of Orange and Green Lines:** The orange line (`⟨M⟩_0^{L(4)}|ψ_t^g⟩`) and the green line (`⟨ψ_0^g|M|ψ_0^g⟩`) track each other very closely for high `F` values across all `p`. Their separation is a key indicator of the breakdown of an approximation or the influence of the parameter `p` at lower `F`. 4. **Systematic Offset:** For `p=0.50` and `F ≈ 0.40`, the orange line is systematically less negative than the green line for `g > 0.25`, suggesting the theoretical approximation (`L(4)`) overestimates the expectation value compared to the initial state calculation under these conditions. ### Interpretation This chart likely comes from a study in quantum many-body physics or quantum simulation, comparing different methods for calculating the expectation value of an observable `M` as a function of a coupling strength `g`. * **Parameter `F`:** This likely represents a **fidelity** or a measure of state quality/purity. A high `F` (~0.99) indicates a high-quality, low-noise state or simulation, leading to smooth, reliable results. A low `F` (~0.40) indicates a noisy, low-fidelity state, causing the direct measurement (blue line) to become unreliable and fluctuate wildly. * **Parameter `p`:** This could represent a **noise rate**, **error probability**, or a **perturbation strength**. Increasing `p` introduces more disturbance into the system, causing the different theoretical and computational approaches to diverge. * **The Three Lines:** * The **blue line** is the "direct" result from the (possibly noisy) time-evolved state. Its noise at low `F` reflects the instability of this direct calculation. * The **green line** is a baseline calculation from the initial state `ψ_0^g`. * The **orange line** appears to be a **theoretical approximation** (e.g., from perturbation theory or a linked-cluster expansion) intended to predict the expectation value for the state `ψ_t^g`. * **Core Finding:** The data suggests that the theoretical approximation (orange) is excellent when the system fidelity `F` is high, as it matches both the direct calculation (blue) and the initial state baseline (green). However, as fidelity `F` decreases and noise `p` increases, the approximation breaks down. It no longer matches the noisy direct calculation and, more importantly, deviates from the initial state baseline, indicating a fundamental change in the system's behavior or the failure of the approximation under noisy, low-fidelity conditions. The chart effectively maps the "phase space" of validity for the theoretical model. </details> Figure 2: Staggered magnetisation for a 6-site 1D Heisenberg antiferromagnet with staggered external field, as a function of field parameter $g$ in the Hamiltonian (Eq. 11). Comparison of methods for obtaining ground state staggered magnetisation – direct $\langle M\rangle$ (blue) and QCM $\langle M\rangle_{0}^{\rm L(4)}$ (orange) estimates vs. exact value (green). To compute the estimates, trial states $\ket{\psi_{t}^{g}}$ were generated at each value of $g$ by starting at the ground state $\ket{\psi_{0}^{g}}$ and applying a random rotation as $\ket{\psi_{t}^{g}}=e^{-i\theta M}\ket{\psi_{0}^{g}}$ , where $M$ is a random Hermitian matrix sampled from the Gaussian unitary ensemble. By tuning the parameter $\theta$ , trial states of varying degrees of overlap with the ground state $F=\left|\langle\psi_{t}^{g}|\psi_{0}^{g}\rangle\right|^{2}$ were generated. These trial states ${\rho}=\ket{\psi_{t}^{g}}\!\bra{\psi_{t}^{g}}$ were then passed through a depolarising error channel ${\rho}\mapsto(1-\frac{3p}{4}){\rho}+\frac{p}{4}(X{\rho}X+Y{\rho}Y+Z{\rho}Z)$ with noise parameter $p$ , which was varied from 1% up to 50%. Of particular interest in the study of quantum magnetism is the Heisenberg XXZ model, given by the Hamiltonian: $$ H=\frac{1}{4q}\sum_{\langle i\,j\rangle}\left(Z_{i}Z_{j}+x(X_{i}X_{j}+Y_{i}Y_{ j})\right). \tag{9} $$ Here, $q$ is the number of qubits, the $S_{i}$ are the Pauli operators $S\in\{X,Y,Z\}$ acting on the $i^{\rm th}$ qubit, the $\langle i\,j\rangle$ are the nearest neighbour couplings in the lattice. The parameter $x$ controls the strength of the transverse spin components relative to the Ising-like interaction. As a first test case of the moments-based ground state observable estimate (Eq. 8), we take the XXZ model over a 2D square lattice, and consider the $ZZ$ correlation operator, $$ C_{ij}=Z_{i}Z_{j}, \tag{10} $$ acting on a pair of next-nearest-neighbour qubits $i$ and $j$ . Focusing on these longer range interactions provides additional insights into the system behaviour beyond what is captured explicitly by the Hamiltonian. On a quantum computer, to simulate and study the ground state behaviour (e.g. the energy and spin-spin correlations) of this system as the parameter $x$ varies, one might need to prepare the ground state using a different circuit at each value of $x$ . However, one can perform far fewer computations by leveraging the fact that the QCM approach has reduced sensitivity to the choice of trial state. A well-chosen state that accurately describes the solution at a given value of $x$ will also produce an accurate QCM estimate for nearby values of $x$ . Figure 1 (a) shows that the ground state energy of the Heisenberg XXZ model can be fully characterised for all $-1<x<1$ using the QCM estimate $E_{0}^{\rm L(4)}$ with respect to just three different trial states (ground states at $x\in\{-1,0,1\}$ ), in stark contrast to $\langle H\rangle$ from the standard variational approach. In Figure 1 (b), the difference between the two methods is made even clearer when these states are used to find estimates of other ground state properties. QCM correctly estimates the correlation in the ground state between the corner site and its next-nearest-neighbour in the middle of the $4\times 3$ square lattice, remaining remarkably close to the exact value as $x$ varies, diverging only slightly at $x=\pm\frac{1}{2}$ where the ground state overlap is at a minimum. In practice on a real device, measuring the moments up to $H^{4}$ requires one to multiply out the terms (or Pauli strings) in the Hamiltonian and determine their expectation values. These Pauli strings can be grouped into simultaneously measurable tensor product basis (TPB) sets, the total number of which scales logarithmically with the number of Pauli strings [12]. Each TPB corresponds to one measurement on the quantum computer – thus when determining a general ground state observable using QCM, if the operator describes interactions present explicitly in the Hamiltonian, no additional measurements need to be made over the energy calculation. In our Heisenberg XXZ case, to go from the energy calculation in Figure 1 (a) to the next-nearest-neighbour correlation in Figure 1 (b), the total number of Pauli strings increases from 66 343 to 68 960, and the total number of TPB measurements goes from 1 906 to 1 973. Note that these measurements need to only be made once for each of the three trial states, after which the entire curve for $-1<x<1$ can be filled out via simple classical post-processing. So far, we have shown that the QCM estimate of a ground state observable other than energy shows the same resilience to an imperfect choice of trial state, however the presence of the noise robustness property still remains to be verified. For this, we turn our attention to the antiferromagnetic Heisenberg model with an external (staggered) field: $$ H=\frac{1}{4}\sum_{\langle i\,j\rangle}(X_{i}X_{j}+Y_{i}Y_{j}+Z_{i}Z_{j})+ \frac{g}{2}\sum_{k}(-1)^{k}Z_{k}, \tag{11} $$ where $g$ is the external field parameter. An operator of interest in this case is the staggered magnetisation $M$ , given by: $$ M=\frac{1}{2}\sum_{k}(-1)^{k}Z_{k}. \tag{12} $$ We look at this model over 6 qubits in 1D. To simulate the results one might typically see in a variational quantum algorithm, given that preparing ansaẗze that can fully represent the target state is often challenging, we approximate an intermediate step of such an algorithm by starting with the exact ground state vector $\ket{\psi_{0}^{g}}$ for each $g$ and applying random rotations to obtain a “trial state” $\ket{\psi_{t}^{g}}$ . Both the direct and moments-based estimates of the staggered magnetisation are then calculated with respect to this trial state after applying various levels of depolarising noise. Figure 2 shows the interplay between these two phenomena in our simulations for varying average ground state fidelity, $F=\left|\langle\psi_{t}^{g}|\psi_{0}^{g}\rangle\right|^{2}$ , and depolarising noise, ${\rho}\mapsto(1-\frac{3p}{4}){\rho}+\frac{p}{4}(X{\rho}X+Y{\rho}Y+Z{\rho}Z)$ with parameter $p$ . We observe that the robustness of the QCM technique to low fidelity trial states and noise, previously only seen in the context of energy problems [12, 13, 14], persists when considering arbitrary ground state observables. We see even for trial states with $\sim$ 40% ground state fidelity and 50% depolarising error, at each value of $g$ , the moments based estimate $\langle M\rangle_{0}^{\rm L(4)}$ provides a powerful correction to the direct measurement of $\langle M\rangle$ , remaining remarkably close to the exact value for the ground state staggered magnetisation. This is more or less the level of noise one would expect to see in a typical variational circuit on real hardware with 1% CNOT error, so the result bodes well for the prospect of applying this technique on present-day devices. ## IV Conclusion We have presented an extension of the QCM method to estimate arbitrary ground state observables of quantum systems, with a focus on the Heisenberg model. We observe the robustness of the QCM technique to low fidelity trial states and noise, previously only seen in the context of ground-state energy problems. In particular, the results presented in this paper demonstrate the ability of QCM to estimate ground state properties such as magnetisation and spin-spin correlations of the Heisenberg model to a high precision, even with suboptimal trial states and levels of noise present in near-term quantum hardware. Further advancement of this technique will lead to improved understanding of the conditions under which various QCM observable estimates hold greater or lesser validity. We reiterate that the results presented here are from simplified noisy simulations, and the key next steps will involve performing the technique on real quantum hardware and under improved noise models – previous analysis of the QCM energy estimate in [13] in addition to the preliminary results presented here for the generalisation to arbitrary observables suggests that we should also expect this technique to work quite well on a real noisy device. Furthermore, the examples we have studied here have only scratched the surface of possible Hamiltonians and observables that could be suitable for this approach. Many intriguing problems invite future application of this technique, such as calculating ground state observables for the Fermi-Hubbard model [22] and detecting phase transitions in more unconventional systems by measuring nonlocal order parameters [23]. QCM has so far proven to be an exciting and potentially useful technique for obtaining the ground-state of many body systems on quantum computers, even in the presence of noise and with suboptimal trial states. Our findings here generalise the method to other observables of interest and further highlight its practical potential for solving a wide range of problems on near-term quantum hardware. ## Acknowledgements This research was supported by the University of Melbourne through the establishment of the IBM Quantum Network Hub at the University. 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