What Do You Mean “Simulating a Quantum Computation?” David Poulin IQC, University of Waterloo & Perimeter Institute October 2002

A second example of what Chris called “a bad question”. In condensed matter physics, it is often quite useful to introduce the notion of “quasi-particles”. These are excitations which behave almost like free particles but have extra weird features. For example, the mass of the quasi-particle may depend on the direction of its motion: a mass tensor. Yet, people don’t organize meetings on the interpretation of quasi-particles! David Poulin, IQC University of Waterloo & PI

It is clear at this time that quantum mechanics is not the final theory. In whatever turns out to be the final theory (string theory, quantum loop gravity, etc.), quantum mechanics will only be a good approximation. It is also possible that some of the weirdness disappears. But we are here having this meeting! 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 think about simulation?

David Poulin, IQC University of Waterloo & PI QS QCCC 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 with CC “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 good as the simulated one! How much of the stuff on the previous slide can we measure 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 simulation of QS is a hard problem is Gauss’ failure at finding an efficient 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 about simulating probability.” ---Feynman There are two ways of addressing this problem: 1. Simulate the “wave packet dynamics”  (x,t) like one would do with water waves. 2.Use a probabilistic CC which “reproduces some statistical properties of the system”. 1. 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 & Linden p-blockness: 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. Entanglement is only related to simulatability through the way we chose to represent the wave function.

David Poulin, IQC University of Waterloo & PI If we insist on computing an exponential amount of extra unphysical information (  ), the exponential overhead is inevitable. Slightly weaker notion of “simulating probabilities”: Reproduce the probabilities of a fixed final measurement. Inputs: I = {G i } Outputs: O = {H j } G i H j  p ij QM Reproduce p ij for all choice of {H j }   “Unperformed experiments have no results” ---Peres

David Poulin, IQC University of Waterloo & PI Simulate physics, not counterfactual experiments p-blockness  p-blockability! F = { I, Q 1, Q 2,..., Q L, O } Q k = p-block states L is the circuit’s depth If F form a family of consistent histories, then the measurements Q k can be carried out --- collapsing the state to a p-block state --- without changing the factual (physically meaningful) probabilities p ij.

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 “statistically reproduce 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 = 2 50 << Number of molecules = /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 to classical 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!