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Quasi-quantum states and the quasi-quantum PCP theorem

Itai Arad, Miklos Santha

TL;DR

This work defines $k$-local quasi-quantum (qq) states as a convex relaxation of quantum states achieved by a SIC-POVM map, yielding a classical distributional perspective on $k$-local Hamiltonians. It shows that optimizing the $k$-local Hamiltonian over qq states is NP-hard and provides a PCP-style hardness-of-approximation result, offering a tractable model that mirrors certain quantum LH complexities. Central innovations include the scapegoat mechanism to concentrate energy penalties, the introduction of $\lambda$-solutions on $3$-decomposable graphs, and a detailed reduction from 4-coloring to a $3$-local qq Hamiltonian with a large promise gap. The results offer insights into quantum PCP conjectures by illustrating potential gap-amplification obstacles and by suggesting a route to quasi-quantum analogs of PCP reductions, while also proposing a practical optimization framework for quantum Hamiltonians via qq-state relaxations.

Abstract

We introduce $k$-local quasi-quantum states: a superset of the regular quantum states, defined by relaxing the positivity constraint. We show that a $k$-local quasi-quantum state on $n$ qubits can be 1-1 mapped to a distribution of assignments over $n$ variables with an alphabet of size $4$, which is subject to non-linear constraints over its $k$-local marginals. Therefore, solving the $k$-local Hamiltonian over the quasi-quantum states is equivalent to optimizing a distribution of assignment over a classical $k$-local CSP. We show that this optimization problem is essentially classical by proving it is NP-complete. Crucially, just as ordinary quantum states, these distributions lack a simple tensor-product structure and are therefore not determined straightforwardly by their local marginals. Consequently, our classical optimization problem shares some unique aspects of Hamiltonian complexity: it lacks an easy search-to-decision reduction, and it is not clear that its 1D version can be solved with dynamical programming (i.e., it could remain NP-hard). Our main result is a PCP theorem for the $k$-local Hamiltonian over the quasi-quantum states in the form of a hardness-of-approximation result. The proof suggests the existence of a subtle promise-gap amplification procedure in a model that shares many similarities with the quantum local Hamiltonian problem, thereby providing insights on the quantum PCP conjecture.

Quasi-quantum states and the quasi-quantum PCP theorem

TL;DR

This work defines -local quasi-quantum (qq) states as a convex relaxation of quantum states achieved by a SIC-POVM map, yielding a classical distributional perspective on -local Hamiltonians. It shows that optimizing the -local Hamiltonian over qq states is NP-hard and provides a PCP-style hardness-of-approximation result, offering a tractable model that mirrors certain quantum LH complexities. Central innovations include the scapegoat mechanism to concentrate energy penalties, the introduction of -solutions on -decomposable graphs, and a detailed reduction from 4-coloring to a -local qq Hamiltonian with a large promise gap. The results offer insights into quantum PCP conjectures by illustrating potential gap-amplification obstacles and by suggesting a route to quasi-quantum analogs of PCP reductions, while also proposing a practical optimization framework for quantum Hamiltonians via qq-state relaxations.

Abstract

We introduce -local quasi-quantum states: a superset of the regular quantum states, defined by relaxing the positivity constraint. We show that a -local quasi-quantum state on qubits can be 1-1 mapped to a distribution of assignments over variables with an alphabet of size , which is subject to non-linear constraints over its -local marginals. Therefore, solving the -local Hamiltonian over the quasi-quantum states is equivalent to optimizing a distribution of assignment over a classical -local CSP. We show that this optimization problem is essentially classical by proving it is NP-complete. Crucially, just as ordinary quantum states, these distributions lack a simple tensor-product structure and are therefore not determined straightforwardly by their local marginals. Consequently, our classical optimization problem shares some unique aspects of Hamiltonian complexity: it lacks an easy search-to-decision reduction, and it is not clear that its 1D version can be solved with dynamical programming (i.e., it could remain NP-hard). Our main result is a PCP theorem for the -local Hamiltonian over the quasi-quantum states in the form of a hardness-of-approximation result. The proof suggests the existence of a subtle promise-gap amplification procedure in a model that shares many similarities with the quantum local Hamiltonian problem, thereby providing insights on the quantum PCP conjecture.

Paper Structure

This paper contains 27 sections, 12 theorems, 47 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Quantum vs. classical constraint satisfaction problems
  3. Quasi-quantum states
  4. Overview of the proof
  5. Discussion
  6. Background
  7. Quantum states and SIC-POVM measurements
  8. k-local quasi-quantum states (k-qq states)
  9. Definition of k-local quasi-quantum states
  10. The support of a k-qq state
  11. The k-local Hamiltonian problem over quasi-quantum states
  12. The quasi-quantum PCP theorem
  13. The 4-coloring problem over qq states
  14. The scapegoat mechanism
  15. lambda-solutions and 3-decomposable graphs
  16. ...and 12 more sections

Key Result

Theorem 1.1

Consider the following problem. We are given a $3$-local Hamiltonian $H=\sum_i h_i$ together with two numbers $a<b$ and a promise that either $\epsilon_0^{qq}\le a$ ( YES case) or $\epsilon_0^{qq}\ge b$ ( NO case), where $\epsilon_0^{qq}$ is the quasi-quantum ground energy over $3$-local quasi-quant

Figures (6)

  • Figure 1: A schematic representation of the relations between quantum states, $k$-local qq states and general probability distributions. The $k$-local qq states are a superset of the quantum states, and are a convex subset of the polytope of all possible probability distributions of assignments over the alphabet $\{0,1,2,3\}^n$.
  • Figure 2: An illustration of an unsatisfiable classical CSP, where every variable $x_i$ takes value in $\{0,1\}$, and on every edge we place a constraint that is satisfied only if its adjacent variables take different values.
  • Figure 3: A common choice for the $4$ elements of a SIC-POVM of qubit (see Eq. (\ref{['eq:POVM-d2']})) form a tetrahedral on the Bloch sphere.
  • Figure 4: Illustration of the construction of the $H_G$ Hamiltonian, given in \ref{['def:H']}. (a) We start with a 3-decomposable graph, where the edges have been partitioned into $3$ subsets, red, greed, blue. (b) We place qubits on the vertices of the graph and add $3$ scapegoat qubits to the system, one for each subset. The terms in the local Hamiltonian are now $3$-local: there are coloring terms, which involve the qubits of an edge, together with its associated scapegoat, and there is the scapegoat constraint, which involves all three scapegoats.
  • Figure 13: The construction of a $\lambda$-solution for cycles. We index the vertices of the cycle by $\{1,2,\ldots, n\}$ and mark the even and odd vertices differently. When the cycle has an odd number of vertices, the two odd vertices $1,n$ are adjacent.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1: SIC-POVM
  • Definition 2.2: Dual basis of a SIC-POVM
  • Lemma 2.3
  • Definition 3.1: $k$-local quasi-quantum states
  • Definition 3.2: An equivalent definition of $k$-local quasi-quantum states
  • Definition 3.3: Support of a qq state
  • Lemma 3.4
  • Theorem 3.5: Sister states
  • Definition 4.1: The approximation scale $L_H$
  • ...and 19 more