• Corpus ID: 222140868

StoqMA vs. MA: the power of error reduction

  title={StoqMA vs. MA: the power of error reduction},
  author={Dorit Aharonov and Alex Bredariol Grilo and Yupan Liu},
StoqMA characterizes the computational hardness of stoquastic local Hamiltonians, which is a family of Hamiltonians that does not suffer from the sign problem. Although error reduction is commonplace for many complexity classes, such as BPP, BQP, MA, QMA, etc.,this property remains open for StoqMA since Bravyi, Bessen and Terhal defined this class in 2006. In this note, we show that error reduction forStoqMA will imply that StoqMA = MA. 
4 Citations
StoqMA meets distribution testing
  • Yupan Liu
  • Mathematics, Computer Science
  • 2021
This work proves that easy-witness $\mathsf{StoqMA}$ is a generalization of a subset state such that the associated set's membership can be efficiently verifiable, and all non-zero coordinates are not necessarily uniform, and makes a step towards collapsing the hierarchy.
The Complexity of Translationally Invariant Problems beyond Ground State Energies
This work shows that the translationally invariant versions of both APX-SIM and GSCON remain intractable, and gives a framework for lifting any abstract local circuit-to-Hamiltonian mapping $H$ (satisfying mild assumptions) to hardness of APx-SIM on the family of Hamiltonians produced by $H$, while preserving the structural and geometric properties of $H $ (e.g. translation invariance, geometry, locality, etc).
On polynomially many queries to NP or QMA oracles
This work shows that for any verification class C, any P C machine with a query graph of “separator number” s can be simulated using deterministic time exp, and shows how to combine Gottlob’s “admissible-weighting function’ framework with the “flag-qubit” framework for embedding P C computations directly into APX-SIM instances in a black-box fashion.
Termwise versus globally stoquastic local Hamiltonians: questions of complexity and sign-curing
We elucidate the distinction between global and termwise stoquasticity for local Hamiltonians and prove several complexity results. We show that the stoquastic local Hamiltonian problem is


Stoquastic PCP vs. Randomness
  • D. Aharonov, A. B. Grilo
  • Mathematics
    2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2019
This work shows that the gapped version of this problem, i.e. deciding if a given uniform stoquastic local Hamiltonian is frustration-free or has energy at least some constant ε, is in NP, and reveals a new MA-complete problem, stated in terms of classical constraints on strings of bits, which is the first (arguably) natural MA- complete problem stated in non-quantum CSP language.
The complexity of stoquastic local Hamiltonian problems
It is proved that LH-MIN for stoquastic Hamiltonians belongs to the complexity class AM -- a probabilistic version of NP with two rounds of communication between the prover and the verifier, and that any problem solved by adiabatic quantum computation using stoquian Hamiltonians is in PostBPP.
Merlin-Arthur Games and Stoquastic Complexity
It is proved that MA has a natural complete problem which is called the stoquastic k-SAT problem, a matrix-valued analogue of the satisfiability problem in which clauses are k-qubit projectors with non-negative matrix elements, while a satisfying assignment is a vector that belongs to the space spanned by these projectors.
Achieving perfect completeness in classical-witness quantum merlin-arthur proof systems
This paper proves that classical-witness quantum Merlin-Arthur proof systems can achieve perfect completeness. That is, QCMA = QCMA1. This holds under any gate set with which the Hadamard and
Complexity Classification of Local Hamiltonian Problems
This work characterises the complexity of the k-local Hamiltonian problem for all 2-local qubit Hamiltonians and proves for the first time QMA-completeness of the Heisenberg and XY interactions in this setting.
Complexity of Stoquastic Frustration-Free Hamiltonians
The Cook-Levin theorem proving NP-completeness of the satisfiability problem is generalized to the complexity class MA (Merlin-Arthur games)—a probabilistic analogue of NP.
Two Combinatorial MA-Complete Problems
Two new combinatorial problems are defined and proved to be MA-completeness: ACAC, which gets as input a succinctly described graph, with some marked vertices, and SetCSP, which generalizes standard constraint satisfaction problem (CSP) into constraints involving sets of strings.
The Complexity of the Local Hamiltonian Problem
This paper settles the question and shows that the 2-LOCAL HAMILTONIAN problem is QMA-complete, and demonstrates that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.
On Complexity of the Quantum Ising Model
We study complexity of several problems related to the Transverse field Ising Model (TIM). First, we consider the problem of estimating the ground state energy known as the Local Hamiltonian Problem
Two-local qubit Hamiltonians: when are they stoquastic?
We examine the problem of determining if a 2-local Hamiltonian is stoquastic by local basis changes. We analyze this problem for two-qubit Hamiltonians, presenting some basic tools and giving a