Despite immense progress in quantum Hamiltonian complexity in the past decade, little is known about the computational complexity of quantum physics at the thermodynamic limit. In fact, even defining the problem properly is not straight forward. We study the complexity of estimating the ground energy of a fixed, translationally-invariant (TI) Hamiltonian in the thermodynamic limit, to within a given precision; this precision (given by n the number of bits of the approximation) is the sole input to the problem. Understanding the complexity of this problem captures how difficult it is for a physicist to measure or compute another digit in the approximation of a physical quantity in the thermodynamic limit. We show that this problem is contained in FEXPQMA-EXP and is hard for FEXPNEXP. This means that the problem is doubly exponentially hard in the size of the input. As an ingredient in our construction, we study the problem of computing the ground energy of translationally invariant finite 1D chains. A single Hamiltonian term, which is a fixed parameter of the problem, is applied to every pair of particles in a finite chain. In the finite case, the length of the chain is the sole input to the problem and the task is to compute an approximation of the ground energy. No thresholds are provided as in the standard formulation of the local Hamiltonian problem. We show that this problem is contained in FPQMA-EXP and is hard for FPNEXP. Our techniques employ a circular clock structure in which the ground energy is calibrated by the length of the cycle. This requires more precise expressions for the ground states of the resulting matrices than was required for previous QMA-completeness constructions and even exact analytical bounds for the infinite case which we derive using techniques from spectral graph theory. To our knowledge, this is the first use of the circuit-to-Hamiltonian construction which shows hardness for a function class.
|Original language||American English|
|Title of host publication||STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing|
|Editors||Stefano Leonardi, Anupam Gupta|
|Publisher||Association for Computing Machinery|
|Number of pages||14|
|State||Published - 6 Sep 2022|
|Event||54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 - Rome, Italy|
Duration: 20 Jun 2022 → 24 Jun 2022
|Name||Proceedings of the Annual ACM Symposium on Theory of Computing|
|Conference||54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022|
|Period||20/06/22 → 24/06/22|
Bibliographical notePublisher Copyright:
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- Hamiltonian Complexity
- Thermodynamic Limit