Quantum control without quantum states

TL;DR

This work develops an operator-centered quantum-control framework that avoids explicit state-vector propagation by evolving quantum invariants $\mathcal{I}(t)$ via $\partial_t\mathcal{I}+i[H(t),\mathcal{I}]=0$. By expanding $\mathcal{I}(t)$ in a closed operator basis and propagating the coefficient vector $\mathbf{a}(t)$ with $\dot{\mathbf{a}}(t)=K(t)\mathbf{a}(t)$, the approach can achieve control tasks with polynomial rather than exponential scaling in system size for suitable algebras. The authors demonstrate state preparation for GHZ and cluster states, and propagator realization, on a driven spin chain with extended Ising interactions, obtaining operator infidelities around $10^{-5}$ up to $n=50$ spins, and describe broader families of operator sets with similar scaling. The method, compatible with existing gradient-based optimizers, offers a scalable route to quantum-state engineering and the realization of higher-body effective dynamics, with potential impact on quantum simulation and error-correcting codes. Overall, the paper provides explicit constructions and numerical evidence that invariant-based, implicit control can extend controllable regimes beyond what traditional state-based methods can feasibly address.

Abstract

We show that combining ideas from the fields of quantum invariants and of optimal control can be used to design optimal quantum control solutions without explicit reference to quantum states. The states are specified only implicitly in terms of operators to which they are eigenstates. The scaling in numerical effort of the resultant approach is not given by the typically exponentially growing effort required for the specification of a time-evolved quantum state, but it is given by the effort required for the specification of a time-evolved operator. For certain Hamiltonians, this effort can be polynomial in the system size. We describe how control problems for state preparation and the realization of propagators can be formulated in this approach, and we provide explicit control solutions for a spin chain with an extended Ising Hamiltonian. The states considered for state-preparation protocols include eigenstates of Hamiltonians with more than pairwise interactions, and these Hamiltonians are also used for the definition of target propagators. The cost of describing suitable time-evolving operators grows only quadratically with the system size, allowing us to construct explicit control solutions for up to 50 spins. While sub-exponential scaling is obtained only in special cases, we provide several examples that demonstrate favourable scaling beyond the extended Ising model.

Figures (7)

• Figure 1: Linear spin chain of $n$ qubits with nearest-neighbor $XX$ interactions. Single-qubit $Z$-driving and $X$-driving on the end qubits enables efficient preparation of various quantum states, including GHZ and cluster states, and the realization of effective dynamics with multi-qubit interactions.
• Figure 2: Infidelity for dynamics for a spin chain (Eq. \ref{['eq:ham']}) with optimized single-qubit driving, as function of systems size $n$. Solid shapes depict operator infidelities $\cal J$ (Eq. \ref{['eq:J']}) in log-scale for state transfer (panel a) and operator infidelities ${\cal J}_{U_T}$ (Eq. \ref{['eq:fidgate']}) for quantum gates (panel b). Empty diamonds and squares in panel a depict bounds (Eq. \ref{['bound_infidelity']}) on state infidelities; empty triangles in panel a represent state infidelities based on matrix-product-state representations and empty shapes in panel b depict gate infidelities. As result of the optimization, the values of the infidelities $\cal J$ and ${\cal J}_{U_T}$ are largely independent of the system size $n$. Actual state- and gate-infidelities do grow with increasing system size, but they remain well below what can be achieved given the experimental imperfections of actual quantum devices.
• Figure 3: Optimised pulses, $f_1$, $f_{25}$ and $f_{50}$, for state transfer in a chain with 50 spins as function of time $t$ in multiples of the interaction time $\tau_g=2\pi/g$. The operator infidelities obtained with these pulses are $\approx 10^{-5}$ for the GHZ state (a), $\approx 10^{-5}$ for the cluster state (b), and $\approx 2\times 10^{-5}$ for the ground state of $H_D$ (c). The duration of controlled dynamics is longer for $H_D$ in (c) than in the other cases. This is likely due to the difficulty in realising the ground state of a gap-less model.
• Figure 4: Optimised pulses, $f_1$, $f_{25}$ and $f_{50}$, for the realization of propagators in a chain with 50 spins as function of time $t$ in unit of $\tau_g$. The resultant operator infidelities are $\approx 4 \times 10^{-5}$ for $U_C$ (a), $\approx 9\times 10^{-5}$ for $U_D$ (b), and $\approx 10^{-4}$ for $U_G$ (c). For $U_G$$f_{50}=f_1$ and hence is not shown. The pulses for $U_G$ in (f) are longer due to the difficulty in quantum-simulating an $n$-body interaction with a pairwise controlled Hamiltonian.
• Figure 5: Spin comb with nearest-neighbor $XY$ interactions in the top chain and $ZZ$ interactions between spins of different chains (inset (a)). Single-spin $X$-driving (inset (a)) enables the realization of effective dynamics with four-spin interactions matching the Toric Code Plaquette operators Eq. \ref{['eq:plaquette']} (inset (b)).