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What Do You Mean “Simulating a Quantum Computation?”

David PoulinIQC, University of Waterloo

&Perimeter Institute

November 2002

David Poulin, IQC University of Waterloo & PI

What Do You Mean “Simulating a Quantum Computation?”

How is this simulation business related to foundation of QM?

“A journey from ontic to epistemic ... with consequences”

Does this have consequences on the way we thinkabout simulation?

David Poulin, IQC University of Waterloo & PI

QS

QC CC

Outline

What is known

• Some QSs can be simulated efficiently on a QC.• “Simulating the dynamics” of some QS is as hard as factoring.• Entanglement is necessary for Q-computational speed-up with pure states.• Finding the ground state of a QS can be NP complete.• etc.

David Poulin, IQC University of Waterloo & PI

Stuff about QS we usually compute withCC “simulations” (at an exponential cost).

• Ground state energy• Properties of the thermal/ground state (symmetries)• Propagators • Degeneracy of energy levels• Transport properties• Properties of spectral functions• Properties of cross section• Partition function• etc.

)(

David Poulin, IQC University of Waterloo & PI

The real thing should be at least as goodas the simulated one!

How much of the stuff on the previous slide can wemeasure from the QS itself... ... or a polynomial number of copies of it?

Does there exist physical quantities extractable from poly copies of a QS which requires exponential CC?

“The strongest argument indicating that the simulationof QS is a hard problem is Gauss’ failure at finding anefficient algorithm for factoring.” ---Gilles (maybe in a dream...)

David Poulin, IQC University of Waterloo & PI

“So I know that quantum mechanics seems to involve probability --- and I therefore want to talk aboutsimulating probability.” ---Feynman

There are two ways of addressing this problem:

1. Simulate the “wave packet dynamics” (x,t) likeone would do with water waves.

2. Use a probabilistic CC which “reproduces somestatistical properties of the system”.

211 . Simulating “the factual probabilities”.

David Poulin, IQC University of Waterloo & PI

“One method for classically simulating a quantum computation is to directly compute the state at each step from the sequence of unitary operations prescribed in the quantum algorithm.”

--- Jozsa & Lindenp-blockness: k 2 1

On at most p qubits

Writing the wave function requires complex amplitudes.

Every step of the computation requires at most complex multiplications... must figure out what constitute the new blocs.

22 2 2 pp

pnk

p42

Entanglement is only related to simulatability through theway we chose to represent the wave function.

David Poulin, IQC University of Waterloo & PI

If we insist on computing an exponential amount of extraunphysical information (), the exponential overhead isinevitable.

Slightly weaker notion of “simulating probabilities”:Reproduce the probabilities of a fixed final measurement.

Inputs: I = {Gi} Outputs: O = {Hj}

Gi Hj pijQM

Reproduce pij for all choice of {Hj}

“Unperformed experiments have no results”---Peres

David Poulin, IQC University of Waterloo & PI

Simulate physics, not counterfactual experiments

p-blockness p-blockability!

F = {I, Q1 , Q2 , ..., QL , O} Qk = )()()( kj

kj

kj aaA

p-block statesL is the circuit’s depth

If F form a family of consistent histories, then the measurements Qk can be carried out --- collapsing the state to a p-block state --- without changing the factual (physically meaningful) probabilities pij .

David Poulin, IQC University of Waterloo & PI

If it is possible to simulate the “wave packet’s dynamics” or the “factual probabilities” it is possible to “statisticallyreproduce the behavior of the QS”.

... but it seams otherwise impossible!

Probabilistic simulation

Are we being fair with CCs?

Computation: Problems which require exponential resources are intractable.

Physics: Properties which require exponential resources to be estimated are practically not measurable.

David Poulin, IQC University of Waterloo & PI

But Avogadro’s number is so large!

It takes a while before the exponential kicks in.

Ex. Molecule: N = 50 hydrogen-like 2-levels atoms. Sample: m = 1g. Number of states = 250 << Number of molecules = 1024/50(7 orders of magnitude!)

Reproducing the statistics is not a fair requirement...... what about some coarse grained version of it?

Coarse graining leads to consistency... which leads toclassical simulatability!

If N = 100, then m has to be > 1Tonne!!!

David Poulin, IQC University of Waterloo & PI

When asking a CC to simulate a QS, we should only ask about things we can actually measure on that system.

Should we expect more from a QC?

... it’s not completely crazy.

Ex. Is the ground state of this QS degenerated?

Beyond simulating!