As quantum computers grow more capable, simulating them on conventional hardware becomes more challenging yet more attractive since this helps in design and verification. Some quantum algorithms and circuits are amenable to surprisingly efficient simulation, and this makes hard-to-simulate computations particularly valuable. For such circuits, we develop accurate massively-parallel simulation with dramatic speedups over earlier methods on 42- and 45-qubit circuits. We propose two ways to trade circuit fidelity for computational speedups, so as to match the error rate of any quantum computer. Using Google Cloud, we simulate approximate sampling from the output of a circuit with 7 × 8 qubits and depth 42 with fidelity 0.5% at an estimated cost of $35K.

Many problems can be solved by preparing a specific eigenstate of some Hamiltonian $H$. The generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of $H$ for that eigenstate and the cost of evolving with $H$ for some fixed time. The goal of spectral gap amplification is to construct a Hamiltonian $H'$ with the same eigenstate as $H$ but a bigger spectral gap, requiring that constant-time evolutions with $H'$ and $H$ are implemented with nearly the same cost. We show that a quadratic spectral gap amplification is possible when $H$ satisfies a frustration-free property and give $H'$ for these cases. This results in quantum speedups for optimization problems. It also yields improved constructions for adiabatic simulations of quantum circuits and for the preparation of projected entangled pair states, which play an important role in quantum many-body physics. Defining a suitable black-box model, we establish that the quadratic amplification is optimal for frustration-free Hamiltonians and that no spectral gap amplification is possible, in general, if the frustration-free property is removed. A corollary is that finding a similarity transformation between a stoquastic Hamiltonian and the corresponding stochastic matrix is hard in the black-box model, setting limits to the power of some classical methods that simulate quantum adiabatic evolutions.