Simulating quantum dynamics is one of the most important applications of quantum computers. Traditional approaches for quantum simulation involve preparing the full evolved state of the system and then measuring some physical quantity. Here, we present a different and novel approach to quantum simulation that uses a compressed quantum state that we call the “shadow state”. The amplitudes of this shadow state are proportional to the time-dependent expectations of a specific set of operators of interest, and it evolves according to its own Schrödinger equation. This evolution can be simulated on a quantum computer efficiently under broad conditions. Applications of this approach to quantum simulation problems include simulating the dynamics of exponentially large systems of free fermions or free bosons, the latter example recovering a recent algorithm for simulating exponentially many classical harmonic oscillators. These simulations are hard for classical methods and also for traditional quantum approaches, as preparing the full states would require exponential resources. Shadow Hamiltonian simulation can also be extended to simulate expectations of more complex operators such as two-time correlators or Green’s functions, and to study the evolution of operators themselves in the Heisenberg picture. When performing quantum simulation, oftentimes the properties of interest are only a subset of the information contained in the entire state. Here, the authors devise a different type of quantum simulation, where they work with a compressed quantum state whose amplitudes are proportional to expected values of some specific observables of interest.
Hamiltonian truncation offers a nonperturbative route to quantum field theory, yet its accuracy is limited by the rapid expansion of the truncated Hilbert space, which drives up computational cost. We tackle this bottleneck with a hybrid strategy that pairs classical and quantum algorithms: 1) we develop an efficient basis-generation scheme built on integer partitions; 2) we speed up the construction of the sparse Hamiltonian matrix using symmetry-aware algorithms; and 3) we explore quantum Krylov diagonalization as a route to the low-lying spectrum. Benchmarking against the free massive scalar and
φ4 theories in two spacetime dimensions, we achieve substantial gains in the computational efficiency of Hamiltonian truncation and chart a path toward future quantum implementations.
By decomposing quantum dynamics across the Lie orbits of a list of observables, we find polynomial bounded classical simulations for the dynamics of the quantum system given a polynomial sized dynamic Lie algebra (DLA) for the generators of the quantum system. To do so, we describe how to construct the Dynamic Observable Subspace (DOS) that captures all the relevant dynamics for calculating expectation values for a specific observable for Pauli strings, diffusor mixers, and general local generators. Efficient sparse matrix representation, decoupling nonlinearity from such representation and basis construction with permissible pruning allows us to simulate the closed dynamics of such systems with hundreds of qubits. Moreover, we find that while restricted DOS circuits may not express the space of an associated Hamiltonian, they can dramatically outperform a fully expressive circuit due to the absence of the Barren Plateau. While classically simulatable, such circuits can still exhibit a form of quantum advantage through inference, act as warm starting for more expressive circuits, and find high quality trial states for Quantum Amplitude Amplification or Quantum Phase Estimation. In the simulation of quantum dynamics, we find that our restricted circuit can be simulated in polynomial time while producing high quality guiding state for downstream tasks like Quantum Phase Estimation with significantly lower energy than circuits with full expressivity.
We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state
∣ψ⟩, a Hermitian matrix
H, a polynomial
f, and a set of observables
{O1,…,Om}, the goal is to estimate
⟨ψ∣f(H)†Ojf(H)∣ψ⟩ for all
j∈{1,…,m}. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of
f(H). It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary.
We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with
H and
∣ψ⟩ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from
H,
f, and
∣ψ⟩ without prior knowledge of the observables or explicit preparation of the target state proportional to
f(H)∣ψ⟩. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.
Standard variational quantum simulation seeks to reproduce the evolution of the full quantum state, although many applications require only the expectation values of a few observables. We study a variational method for pure-state Hamiltonian dynamics that updates circuit parameters to reproduce the evolution of selected expectation values. An exact error identity guides the choice of observables, motivating a construction based on repeated commutators of the target with the Hamiltonian. For Pauli observables and Pauli-rotation circuits, the update can be estimated without ancillary qubits or controlled operations for overlap estimation. Across six- and ten-qubit spin, fermionic, and molecular benchmarks, the observable-targeted update extends the median time within the target-error tolerance compared with standard variational quantum simulation with equal shot budgets. These results show that directing the variational update toward the target observable can extend accurate simulation without increasing the measurement cost per time step.
We present an algorithm that uses block encoding on a quantum computer to exactly construct a Krylov space, which can be used as the basis for the Lanczos method to estimate extremal eigenvalues of Hamiltonians. While the classical Lanczos method has exponential cost in the system size to represent the Krylov states for quantum systems, our efficient quantum algorithm achieves this in polynomial time and memory. The construction presented is exact in the sense that the resulting Krylov space is identical to that of the Lanczos method, so the only approximation with respect to the exact method is due to finite sample noise. This is possible because, unlike previous quantum Krylov methods, our algorithm does not require simulating real or imaginary time evolution. We provide an explicit error bound for the resulting ground state energy estimate in the presence of noise. For our method to be successful efficiently, the only requirement on the input problem is that the overlap of the initial state with the true ground state must be Ω ( 1 / poly ( n ) ) for n qubits.