An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity class P QMA[log] , and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is P QMA[log] -complete. In this paper, we continue the study of P QMA[log] , obtaining the following results. • The P QMA[log] -completeness result of [Ambainis, CCC 2014] requires O(log n)-local observables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P QMA[log] -complete, resolving an open question of Ambainis.• We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P QMA[log] -complete.• P QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of [Beigel, Hemachandra, Wechsung, SCT 1989] to show P QMA[log] ⊆ PP. This improves the containment QMA ⊆ PP [Kitaev, Watrous, STOC 2000].• A central theme of this work is the subtlety involved in the study of oracle classes in which the oracle solves a promise problem. In this vein, we identify a flaw in [Ambainis, CCC 2014] regarding a P UQMA[log] -hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on [Ambainis, CCC 2014] to obtain P UQMA[log] -hardness for estimating spectral gaps under polynomial-time Turing reductions. . 1 More accurately, QMA is Merlin-Arthur (MA) with a quantum proof and quantum verifier.

One strategy to fit larger problems on NISQ devices is to exploit a tradeoff between circuit width and circuit depth. Unfortunately, this tradeoff still limits the size of tractable problems since the increased depth is often not realizable before noise dominates. Here, we develop qubit-efficient quantum algorithms for entanglement spectroscopy which avoid this tradeoff. In particular, we develop algorithms for computing the trace of the n-th power of the density operator of a quantum system, Tr(ρn), (related to the Rényi entropy of order n) that use fewer qubits than any previous efficient algorithm while achieving similar performance in the presence of noise, thus enabling spectroscopy of larger quantum systems on NISQ devices. Our algorithms, which require a number of qubits independent of n, are variants of previous algorithms with width proportional to n, an asymptotic difference. The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation, allowing us to reuse qubits and increase the circuit depth without suffering the usual noisy consequences. We also introduce the notion of effective circuit depth as a generalization of standard circuit depth suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms and should aid in designing future algorithms. We perform numerical simulations to compare our algorithms to the original variants and show they perform similarly when subjected to noise. Additionally, we experimentally implement one of our qubit-efficient algorithms on the Honeywell System Model H0, estimating Tr(ρn) for larger n than possible with previous algorithms.

No abstract

No abstract