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1. 011011010101001001010100010010110001001010100100100110010010 010010100100101100101010010011001001011011110110001011101011 101000101110010011000101100100101001001110010010011001001001 100111001010010100011010101001001010100100101010011001001001 010010100100101100101010010011001001011011110110101110101001 101000101110010011000101100100101001001110010010011001001001 100111001010001010101010101001001010010010110000010010011010 010010100100101100101010010011001001011011110101010101110101 101000101110010011000101100100101001001110010010011001001001 100111001010001010101001001010110110101011000101001001001110 010010100100101100101010010011001001011011110110000101110101 101000101110010011000101100100101001001110010010011001001001 100111001010001010101010100111101101101010100100110010010010 010010100100101100101010010011001001011011110110101011101011 101000101110010011000101100110110101000100100101100100100101 010010100100101100101010010011001001011011100101010111010111 101000101110010011000101100100101001001110010010011001001001 100111001010001010101010100111000101 01001001111100100100101 100111001010001010101010100111001010100100100010010011101101 What Do You Mean “Simulating a Quantum Computation?” David Poulin IQC, University of Waterloo & Perimeter Institute November 2002

2. What Do You Mean “Simulating a Quantum Computation?” How is this simulation business related to foundation of QM? ... with consequences” Does this have consequences on the way we think about simulation? “A journey from ontic to epistemic David Poulin, IQC University of Waterloo & PI

3. 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. Outline QS QC CC David Poulin, IQC University of Waterloo & PI

4. 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

5. 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

6. 1 . Simulating “the factual probabilities”. “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) likeone would do with water waves. • Use a probabilistic CC which “reproduces somestatistical properties of the system”. David Poulin, IQC University of Waterloo & PI

7. 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. “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 Entanglement is only related to simulatability through the way we chose to represent the wave function. David Poulin, IQC University of Waterloo & PI

8. Inputs: I ={Gi} Outputs: O = {Hj} QM Gi Hj  pij Reproduce pij for all choice of {Hj}   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. “Unperformed experiments have no results” ---Peres David Poulin, IQC University of Waterloo & PI

9. Simulate physics, not counterfactual experiments p-blockness  p-blockability! F = {I, Q1 , Q2, ..., QL , O} Qk = p-block states L is the circuit’s depth If Fform a family of consistent histories, then the measurements Qkcan 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

10. Are we being fair with CCs? Computation: Problems which require exponential resources are intractable. Probabilistic simulation 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! Physics: Properties which require exponential resources to be estimated are practically not measurable. David Poulin, IQC University of Waterloo & PI

11. 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!) If N = 100, then m has to be > 1Tonne!!! 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! David Poulin, IQC University of Waterloo & PI

12. Beyond simulating! 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? David Poulin, IQC University of Waterloo & PI