[Paper Review] Limitations on the simulation of non-sparse hamiltonians

Andrew M. Childs, Robin Kothari|arXiv (Cornell University)|Jul 1, 2010

Quantum Computing Algorithms and Architecture18 references52 citations

TL;DR

This paper establishes fundamental limitations on simulating non-sparse Hamiltonians on quantum computers, generalizing the no-fast-forwarding theorem to dense Hamiltonians and proving that generic simulations cannot run in time o(||Ht||), even when ||H|| is not a reliable size measure. It further rules out poly(||Ht||, log N) time simulations, showing that existing discrete-time quantum walk-based methods cannot be significantly improved in general, though some non-sparse Hamiltonians with low-arboricity graphs can still be simulated efficiently.

ABSTRACT

The problem of simulating sparse Hamiltonians on quantum computers is well studied. The evolution of a sparse N × N Hamiltonian H for time t can be simulated using O(||Ht|| poly(logN)) operations, which is essentially optimal due to a no-fast-forwarding theorem. Here, we consider non-sparse Hamiltonians and show significant limitations ontheir simulation. We generalize the no-fast-forwarding theorem to dense Hamiltonians, ruling out generic simulations taking time o(||Ht||), even though ||H|| is not a uniquemeasure of the size of a dense Hamiltonian H. We also present a stronger limitationruling out the possibility of generic simulations taking time poly(||Ht||; logN), showingthat known simulations based on discrete-time quantum walk cannot be dramatically improved in general. On the positive side, we show that some non-sparse Hamiltonianscan be simulated efficiently, such as those with graphs of small arboricity.

Motivation & Objective

• To investigate the theoretical limits of simulating non-sparse Hamiltonians on quantum computers.
• To extend the no-fast-forwarding theorem from sparse to dense Hamiltonians.
• To determine whether generic simulations of dense Hamiltonians can achieve sub-linear or polynomial dependence on ||Ht|| and log N.
• To identify classes of non-sparse Hamiltonians that remain efficiently simulatable despite general limitations.

Proposed method

• Generalizing the no-fast-forwarding theorem to dense Hamiltonians using spectral norm analysis and quantum query complexity.
• Analyzing the query complexity of quantum algorithms for simulating dense Hamiltonians, focusing on the role of ||Ht|| as a lower bound.
• Using graph-theoretic properties, particularly arboricity, to identify efficiently simulatable non-sparse Hamiltonians.
• Applying discrete-time quantum walk frameworks as a baseline for known simulation techniques, then proving their optimality is unattainable in general.
• Establishing reductions from simulation problems to query problems to derive lower bounds on runtime.

Experimental results

Research questions

• RQ1 Can non-sparse Hamiltonians be simulated in time o(||Ht||), even when ||H|| is not a reliable measure of Hamiltonian size?
• RQ2 Is it possible to achieve generic simulations of dense Hamiltonians in time poly(||Ht||, log N) using current quantum walk-based methods?
• RQ3 What structural properties of a Hamiltonian's interaction graph allow for efficient simulation despite non-sparsity?
• RQ4 How do the limitations on dense Hamiltonian simulation compare to those on sparse Hamiltonians?

Key findings

• The no-fast-forwarding theorem generalizes to dense Hamiltonians, ruling out generic simulations running in time o(||Ht||), even when ||H|| is not a unique measure of size.
• Generic simulations of dense Hamiltonians cannot run in time poly(||Ht||, log N), showing that known discrete-time quantum walk methods cannot be dramatically improved in general.
• The paper establishes a lower bound on the runtime of any quantum algorithm simulating a dense Hamiltonian, based on its spectral norm and evolution time.
• Hamiltonians whose interaction graphs have low arboricity can be simulated efficiently, identifying a broad class of non-sparse Hamiltonians amenable to fast simulation.