Classical shadows for sample-efficient measurements of gauge-invariant observables

Abstract

Classical shadows provide a versatile framework for estimating many properties of quantum states from repeated, randomly chosen measurements without requiring full quantum state tomography. When prior information is available, such as knowledge of symmetries of states and operators, this knowledge can be exploited to significantly improve sample efficiency. In this work, we develop three classical shadow protocols tailored to systems with local (or gauge) symmetries to enable efficient prediction of gauge-invariant observables in lattice gauge theory models which are currently at the forefront of quantum simulation efforts. For such models, our approaches can offer exponential improvements in sample complexity over symmetry-agnostic methods, albeit at the cost of increased circuit complexity. We demonstrate these trade-offs using a $\mathbb{Z}_2$ lattice gauge theory, where a dual formulation enables a rigorous analysis of resource requirements, including both circuit depth and sample complexity.

Figures (10)

• Figure 1: Sample-efficient measurements of gauge-invariant observables in $\mathbb{Z}_2$ LGT. (a) Illustration of $\mathbb{Z}_2$ LGT in $(2+1)$D and its Ising dual. For the LGT qubits reside on links $l$ of the lattice and Gauss law constraints $G_s$ are associated with sites $s$. The Hamiltonian consists of plaquette (magnetic) terms $W_{\square}=\prod_{l\in{\square}}\sigma^z_l$ and on-site (electric) terms $\sigma^x_l$. In the dual description as a $(2+1)$D Ising model the qubit degrees of freedom are associated with the plaquette. There is a one-to-one correspondence between operators and states within the physical Hilbert space of the LGT, independent of boundary conditions: the magnetic terms map to Pauli-$Z$ operators on the Ising qubits and the electric terms map to two-body Pauli-X terms acting on the qubits across the relevant link. In the inverse map from the Ising model to the LGT, single $X$ operators on the Ising side of the duality map to a string of $\sigma^x$ operators along some path $\gamma$ to an arbitrary reference plaquette ${\square}_0$. For PBC, the dual Ising model obeys a global parity constraint and such single-body $X$ operators are unphysical. (b) We introduce three symmetry-aware random measurement protocols—Global and Local Dual Pairs and Dual Product—designed to estimate gauge-invariant observables efficiently, leveraging prior knowledge that expectation values of gauge-variant operators vanish. For Global Dual Pairs, random pairs of Ising qubits are chosen and random two-body symmetry-respecting unitaries associated with these degrees of freedom are mapped back to the LGT side of the duality as the randomizing operations prior to measurement. Local Dual Pairs is similar but leverages a promise of geometric locality in the observables of interest to limit the choice of pairs to local patches. Finally, the Dual Product protocol is constructed from the standard Product Protocol applied to the dual Ising model. This protocol slightly breaks symmetry when we have periodic boundary conditions (PBC) since such operations are not parity-respecting. A trade-off exists between sample complexity and circuit depth. Symmetry-aware protocols offer exponential improvements in sampling efficiency at the cost of increased circuit depth over the standard symmetry-ignorant Product Protocol.
• Figure 2: Schematic overview of the Global Dual Pairs protocol with periodic boundary conditions (PBC).Step 1. A pairing, $\pi$, of dual lattice sites is chosen uniformly at random. For each pair, a random unitary $U_{[ij]}$ is constructed in the Ising formulation. Step 2. For each pair $[ij]\in\pi$, the dual unitary $\mathcal{U}_{[ij]}=\Phi^{-1}[U_{[ij]}]$ is implemented on the LGT side of the duality and a computational basis measurement is performed. Step 3. The output bit string is mapped back to the dual Ising theory, where efficient shadow channel inversion is performed to estimate expectation values of gauge-invariant observables $O$.
• Figure 3: LGT–Ising duality for mapping a (parity-respecting) entangling operation $\Phi^{-1}(Y_i X_j)$. While this operation is two-body on the Ising side (top panels), it corresponds to an extended operation in the LGT (bottom panels) acting between plaquettes $i$ and $j$ along a path $\gamma_{ij}$, which is arbitrary except for its fixed endpoints.
• Figure 4: Overview over the Local Dual Pairs protocol. The Local Dual Pairs protocol is a variant of our scheme for geometrically local gauge-invariant observables that have support on a patch of at most $L\times L$. Step 1 consists of picking a random tiling $\tau\in\mathcal{T}$ and, then, picking a random pairing repeated within each patch of the tiling in order to construct an associated random unitary $U_\pi$. Step 2 consists of applying the associated unitary $\mathcal{U}_\pi$ on the LGT side of the duality, followed by measurement in the computational basis. Step 3 is the post-processing step, which proceeds essentially identically to the Global Dual Pairs protocol restricted to a given patch of a tiling, where the data is filtered to a single choice of tiling according to the support of a given target variable.
• Figure 5: An ancilla enables mixing PBC (even parity) and tPBC (odd parity) sectors. (a) Adding an ancilla $a$ to a reference link $r$ allows us to enforce either PBC or tPBC in the $\mathbb{Z}_{2}$ LGT. (b) In the Dual Product protocol, the $z$-component of the state of the link $r$ is "copied" to the ancilla $a$ via a CNOT. The randomizing unitaries, chosen according to the usual Product Protocol, but applied to the Ising degrees of freedom, then mix between the PBC and tPBC sectors on the LGT side of the duality. In the right panel, we show how the plaquette and Gauss law operators change with the ancilla $a$: The plaquette operators no longer share a link, while the Gauss law operator is effectively extended by a fifth leg, $\sigma^x_r\rightarrow \sigma^x_r \sigma^x_a$.