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The Complexity of Translationally Invariant Problems beyond Ground State Energies

James D. Watson, Johannes Bausch, Sevag Gharibian

TL;DR

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).

Abstract

It is known that three fundamental questions regarding local Hamiltonians -- approximating the ground state energy (the Local Hamiltonian problem), simulating local measurements on the ground space (APX-SIM), and deciding if the low energy space has an energy barrier (GSCON) -- are $\mathsf{QMA}$-hard, $\mathsf{P}^{\mathsf{QMA}[log]}$-hard and $\mathsf{QCMA}$-hard, respectively, meaning they are likely intractable even on a quantum computer. Yet while hardness for the Local Hamiltonian problem is known to hold even for translationally-invariant systems, it is not yet known whether APX-SIM and GSCON remain hard in such "simple" systems. In this work, we show that the translationally invariant versions of both APX-SIM and GSCON remain intractable, namely are $\mathsf{P}^{\mathsf{QMA}_{\mathsf{EXP}}}$- and $\mathsf{QCMA}_{\mathsf{EXP}}$-complete, respectively. Each of these results is attained by giving a respective generic "lifting theorem" for producing hardness results. For APX-SIM, for example, we give 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). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we "compress" the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we give a hardness construction robust against highly non-local unitaries, i.e. even if the adversary acts on all but one qudit in the system in each step.

The Complexity of Translationally Invariant Problems beyond Ground State Energies

TL;DR

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 (satisfying mild assumptions) to hardness of APx-SIM on the family of Hamiltonians produced by , while preserving the structural and geometric properties of (e.g. translation invariance, geometry, locality, etc).

Abstract

It is known that three fundamental questions regarding local Hamiltonians -- approximating the ground state energy (the Local Hamiltonian problem), simulating local measurements on the ground space (APX-SIM), and deciding if the low energy space has an energy barrier (GSCON) -- are -hard, -hard and -hard, respectively, meaning they are likely intractable even on a quantum computer. Yet while hardness for the Local Hamiltonian problem is known to hold even for translationally-invariant systems, it is not yet known whether APX-SIM and GSCON remain hard in such "simple" systems. In this work, we show that the translationally invariant versions of both APX-SIM and GSCON remain intractable, namely are - and -complete, respectively. Each of these results is attained by giving a respective generic "lifting theorem" for producing hardness results. For APX-SIM, for example, we give a framework for "lifting" any abstract local circuit-to-Hamiltonian mapping (satisfying mild assumptions) to hardness of APX-SIM on the family of Hamiltonians produced by , while preserving the structural and geometric properties of (e.g. translation invariance, geometry, locality, etc). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we "compress" the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we give a hardness construction robust against highly non-local unitaries, i.e. even if the adversary acts on all but one qudit in the system in each step.

Paper Structure

This paper contains 48 sections, 26 theorems, 57 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Beyond ground state energies.
  3. Our results.
  4. Hardness of Measuring Local Observables
  5. Hardness of Ground State Connectivity
  6. Techniques
  7. For APX-SIM.
  8. For GSCON.
  9. Paper Structure
  10. Preliminaries
  11. Notation
  12. APX-SIM and GSCON
  13. Approximate simulation.
  14. Ground state connectivity.
  15. Relevant (Oracle) Complexity Classes
  16. ...and 33 more sections

Key Result

Theorem 2

If the family of Hamiltonians $\mathcal{F}$ admits a circuit-to-Hamiltonian mapping such that approximating the ground state energy is $\textup{C}$-hard, then the APX-SIM problem for $\mathcal{F}$ is either $\textup{P}\xspace^{\textup{C[log]}}$ or $\textup{P}\xspace^{\textup{C}}$-complete, depending

Figures (4)

  • Figure 1: The circuit $V$ constructed in \ref{['lem:augmented-circuit']}. The $V_i$ are the Q-verifiers, each taking input $\left\lvert{q_i}\right\rangle$ and proof/witness $\left\lvert{w_i}\right\rangle$. (In principle, states $\left\lvert{w_i}\right\rangle$ can be entangled as one joint state $\left\lvert{w_{1\cdots m}}\right\rangle$; this is dealt with in the proof of \ref{['lem:low-energy-as-many-yes-as-possible']}.) $U'$ denotes the host postprocessing circuit in the original $\textup{D}\xspace\textsuperscript{$\parallel$Q\xspace}$ circuit $U$, which takes the Q-query responses and outputs $U$'s final answer. The gates $R(\sqrt{3}/(2m))$ denote a rotation in the standard basis of angle $\sqrt{3}/(2m)$. For simplicity, we have not depicted any preprocessing needed by $U$ to compute the inputs $\left\lvert{q_i}\right\rangle$ to the Q verifiers $V_i$, nor have we depicted the ancilla register $C$. For clarity, as a black box, the circuit $V$ takes in the input to the $U$ circuit, the joint proof $\left\lvert{w_{1\cdots m}}\right\rangle$, and the ancilla register $C$.
  • Figure 2: (First row) An honest prover essentially counts in unary from left to right, then right to left. (Second row) A cheating prover "shortcuts" this process by using a $b$-local unitary (in the example above, $b=2$) to instantly flip the last $b$ qubits from $0^b$ to $2^b$.
  • Figure 3: The expanded neighbour relations $E$ allowed to overcome Obstacle 3. A ✓ (✗) at position $(i,j)$ means symbol $i$ can (cannot) appear immediately to the left of $j$ in $\mathcal{B}$. For example, $(1,3)$ contains $\text{✗}$, so $E$ forbids substring $13$. Note that $3$ can only have $3$ to its left (column $3$), and $2$ (resp. $4$) can only have $2$ (resp. $4$) to its right (rows $2$ and $4$, respectively).
  • Figure 4: An honest prover's sequence of states in $\mathcal{B}$, organized by phase, and for $N=4$. In this example, for concreteness we assume $F={\left\{1,3\right\}}$.

Theorems & Definitions (61)

  • Definition 1: APX-SIM$(H,A,k,l,a,b,\delta)$ Ambainis2013
  • Theorem 2: LH to APX-SIM (informal)
  • Theorem 3
  • Theorem 4
  • Definition 5: Ground State Connectivity (GSCON) (informal) GS14
  • Theorem 6
  • Definition 7: $\forall$-APX-SIM$(H,A,k,l,a,b,\delta)$ GPY20
  • Definition 8: TI-APX-SIM
  • Definition 9: $\forall$-TI-APX-SIM
  • Definition 10: Ground State Connectivity (GSCON $({H},{k},{\eta_1},{\eta_2},{\eta_3},{\eta_4},{\delta},{b},{m},{U_\psi},{U_\phi})$)GS14
  • ...and 51 more