The complexity of simulating local measurements on quantum systems

Sevag Gharibian1,3 and Justin Yirka2,3

1Department of Computer Science, Paderborn University, Germany
2Department of Computer Science, The University of Texas at Austin, USA
3Department of Computer Science, Virginia Commonwealth University, USA

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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 lower and upper bounds.

Lower bounds (hardness 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 $\textit{single qubit}$ 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.

- 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.

Upper bounds (containment in complexity classes):

- ${P}^{{QMA}[{log}]}$ 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 ${P}^{{QMA}[{log}]}\subseteq {PP}$. This improves the containment ${QMA}\subseteq {PP}$ [Kitaev, Watrous, STOC 2000].

This work contributes a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves a $promise$ problem. This is particularly relevant for quantum complexity theory, where most natural classes such as BQP and QMA are defined as promise classes.

A central question in quantum physics is how many-body systems behave when cooled to near absolute zero. This regime is of particular interest, since it is where phenomena such as superconductivity and superfluidity manifest themselves. If our goal is to understand the $behavior$ of such states, we must probe the properties of the $\textit{ground state}$ of the system, meaning the lowest energy configuration the system relaxes into at low temperature. The most natural avenue for achieving this is to perform local measurements on said ground state, raising the obvious physically motivated question: How difficult is it to simulate such a measurement?

It turns out that the right place to look for the answer is not where the field of quantum Hamiltonian complexity has typically concentrated its effort, Quantum Merlin Arthur (where QMA can be thought of as a “quantum NP”). Rather, Ambainis showed [Ambainis, CCC 2014] that the task of simulating local measurements on ground states is $\textit{even harder than QMA}$, and is characterized by the class PQMA[log] (i.e. the problem is PQMA[log]-complete). Intuitively, this means the problem is believed intractable: the minimal resources to solve it are a classical polynomial-time machine augmented with the “magic” ability to solve a logarithmic number of QMA problems.

The current work continues the study of PQMA[log] and additionally gives a gentle introduction to the topic for the interested reader. We show a variety of results, including (roughly): (1) Showing the problem of simulating measurements on ground states remains PQMA[log]-hard even for more physically motivated settings, such as measurements of just a single qubit of the ground state (as opposed to logarithmically many qubits as in [Ambainis, CCC 2014]). (2) We identify and correct a flaw in the treatment of oracles for promise problems from [Ambainis, CCC 2014] and give a corrected, significantly modified proof of an important result therein, namely for the problem of estimating $\textit{spectral gaps}$ of local Hamiltonians. (3) We show, in a rigorous sense, that while being harder than QMA, PQMA[log] is not $\textit{too much harder}$ (in the formal sense that the latter is still contained in the complexity class PP).

One of the main contributions of this work is a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves a promise problem, such as PQMA[log]. This is particularly relevant for quantum complexity theory, where natural classes such as BQP and QMA are defined as promise classes.

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