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Simulating quantum imaginary-time evolution (QITE) is a major promise ofquantum computation. However, the known algorithms are either probabilistic(repeat until success) with impractically small success probabilities orcoherent (quantum amplitude amplification) but with circuit depths andancillary-qubit numbers unrealistically large in the mid term. Our maincontribution is a new generation of deterministic, high-precision QITEalgorithms significantly more amenable experimentally. These are based on asurprisingly simple idea: partitioning the evolution into several fragmentsthat are sequentially run probabilistically. This causes a huge reduction inwasted circuit depth every time a run fails. Indeed, the resulting overallruntime is asymptotically better than in coherent approaches and the hardwarerequirements even milder than in probabilistic ones, remarkably. Moretechnically, we present two QITE-circuit sub-routines with excellent complexityscalings. One of them is optimal in ancillary-qubit overhead (one singleancillary qubit throughout) whereas the other one is optimal in runtime forsmall inverse temperature or high precision. The latter is shown by noting thatthe runtime saturates a cooling-speed limit that is the imaginary-timecounterpart of the no fast-forwarding theorem of real-time simulations, whichwe prove. Moreover, we also make two technical contributions to the quantumsignal processing formalism for operator-function synthesis (on which oursub-routines are based) that are useful beyond QITE. Our findings are speciallyrelevant for the early fault-tolerance stages of quantum hardware.

Fragmented imaginary-time evolution for early-stage quantum signal processors