Kunal Marwaha

Kunal Marwaha, Bill Fefferman, Alexandru Gheorghiu et al.

We study the complexity of Decoded Quantum Interferometry (DQI), a quantum algorithm for approximate optimization. First, we show that the algorithm resists classical simulation strategies based on locating outputs with large probabilities. We then prove that DQI can be simulated at a low level of the polynomial hierarchy, posing challenges to standard quantum supremacy arguments. We further show that DQI is a constructive solution to a classical coding-theoretic bound based on the MacWilliams identity. Lastly, we interpret DQI as preparing low-energy states of a quantum simple harmonic oscillator, a viewpoint we believe suggests a physics-motivated route to generalizing DQI.

Kunal Marwaha, James Sud

We study the computational complexity of 2-local Hamiltonian problems generated by a positive-weight symmetric interaction term, encompassing many canonical problems in statistical mechanics and optimization. We show these problems belong to one of three complexity phases: QMA-complete, StoqMA-complete, and reducible to a new problem we call EPR*. The phases are physically interpretable, corresponding to the energy level ordering of the local term. The EPR* problem is a simple generalization of the EPR problem of King. Inspired by empirically efficient algorithms for EPR, we conjecture that EPR* is in BPP. If true, this would complete the complexity classification of these problems, and imply EPR* is the transition point between easy and hard local Hamiltonians. Our proofs rely on perturbative gadgets. One simple gadget, when recursed, induces a renormalization-group-like flow on the space of local interaction terms. This gives the correct complexity picture, but does not run in polynomial time. To overcome this, we design a gadget based on a large spin chain, which we analyze via the Jordan-Wigner transformation.