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This figure shows how we can incrementally build the symmetry-protected subspace for U = 2 i=0 iSWAP i,i+1 (θ ). (a) A single unclosed symmetry-protected subspace, or a subgraph of D L for an iSWAP network. All operations L Trot = {L iSWAP 0,1 (θ ) , L iSWAP 1,2 (θ ) , L iSWAP 2,3 (θ ) } which are nonidentity on each node are shown as an edge. Notice the similarities between this graph and the s = 2 subgraph of Fig. 1(a): both have the same vertex set, but the edge set of D L is a subset of the edge set in D U . (b) Follow the recursion relation in Eq. (17) to iteratively build the symmetry-protected subspace G |1100 , starting from |ψ 0 = |1100. Even though the graph in (a) is not equivalent to the graph in Fig. 1(a), their transitively closed graphs are equivalent, as can be seen by the vertices covered by the red line.
Source publication
The analysis of symmetry in quantum systems is of utmost theoretical importance, useful in a variety of applications and experimental settings, and difficult to accomplish in general. Symmetries imply conservation laws, which partition Hilbert space into invariant subspaces of the time-evolution operator, each of which is demarcated according to it...
Contexts in source publication
Context 1
... evolved to by their corresponding unitary operators in O(1) when operating on a single basis state. We recursively build the subspace by checking the set of substring edit maps L ≡ {L U i : U i ∈ U } on each new state, until none are added. This process can be seen as the transitive closure of a subgraph of the graph D L . For U Trot in Eq. (13), Fig. 2(a) shows an example of the subgraph of D L corresponding to the action of each L iSWAP i,i+1 (θ ) ∈ L Trot starting from the initial state |ψ 0 = |1100 (self-edges are ignored as elsewhere in the ...
Context 2
... stop condition activates if no new states are found, that is, when additional operations drawn from L do not unveil any new states. Steps 0-4 in Fig. 2(b) show how the recurrence relation manifests for the input state |ψ 0 = |1100 and the set of substring edit maps generated from the Trotterization in Eq. (13). Once Eq. (17) reaches the stop condition T i+1 |ψ 0 = T i |ψ 0 and returns, we define the symmetry-protected subspace G |ψ 0 to which the state |ψ 0 belongs ...
Citations
... In fact, multiple hopping terms can be measured in parallel in this manner as long as the qubits that the associated Givens rotations affect do not overlap. Measuring the expectation values of Eq. 1 in this way, with only Z-basis and hopping circuits, ensures symmetries are preserved and as an added benefit enables the use of symmetry-based post-selection as an error mitigation technique [51][52][53]. In the absence of measurement circuit parallelization, the number of circuits required to measure the expectation value of Eq. 1 is 1 + 2N imp N bath + N imp (N imp − 1). ...
We present a hardware-reconfigurable ansatz on $N_q$-qubits for the variational preparation of many-body states of the Anderson impurity model (AIM) with $N_{\text{imp}}+N_{\text{bath}}=N_q/2$ sites, which conserves total charge and spin z-component within each variational search subspace. The many-body ground state of the AIM is determined as the minimum over all minima of $O(N_q^2)$ distinct charge-spin sectors. Hamiltonian expectation values are shown to require $\omega(N_q) < N_{\text{meas.}} \leq O(N_{\text{imp}}N_{\text{bath}})$ symmetry-preserving, parallelizable measurement circuits, each amenable to post-selection. To obtain the one-particle impurity Green's function we show how initial Krylov vectors can be computed via mid-circuit measurement and how Lanczos iterations can be computed using the symmetry-preserving ansatz. For a single-impurity Anderson model with a number of bath sites increasing from one to six, we show using numerical emulation that the ease of variational ground-state preparation is suggestive of linear scaling in circuit depth and sub-quartic scaling in optimizer complexity. We therefore expect that, combined with time-dependent methods for Green's function computation, our ansatz provides a useful tool to account for electronic correlations on early fault-tolerant processors. Finally, with a view towards computing real materials properties of interest like magnetic susceptibilities and electron-hole propagators, we provide a straightforward method to compute many-body, time-dependent correlation functions using a combination of time evolution, mid-circuit measurement-conditioned operations, and the Hadamard test.
