Are Quantum States Exponentially Long Vectors? Scott Aaronson (who did and will have an affiliation) (did: IASwill: Waterloo) Distributions over n-bit.

Slides:



Advertisements
Similar presentations
Closed Timelike Curves Make Quantum and Classical Computing Equivalent
Advertisements

Quantum Lower Bound for the Collision Problem Scott Aaronson 1/10/2002 quant-ph/ I was born at the Big Bang. Cool! We have the same birthday.
Quantum Lower Bounds The Polynomial and Adversary Methods Scott Aaronson September 14, 2001 Prelim Exam Talk.
The Power of Unentanglement
How Much Information Is In A Quantum State? Scott Aaronson MIT.
How Much Information Is In Entangled Quantum States? Scott Aaronson MIT |
The Learnability of Quantum States Scott Aaronson University of Waterloo.
Quantum t-designs: t-wise independence in the quantum world Andris Ambainis, Joseph Emerson IQC, University of Waterloo.
Quantum Versus Classical Proofs and Advice Scott Aaronson Waterloo MIT Greg Kuperberg UC Davis | x {0,1} n ?
Quantum Software Copy-Protection Scott Aaronson (MIT) |
The Future (and Past) of Quantum Lower Bounds by Polynomials Scott Aaronson UC Berkeley.
The Learnability of Quantum States Scott Aaronson University of Waterloo.
The Power of Quantum Advice Scott Aaronson Andrew Drucker.
Lower Bounds for Local Search by Quantum Arguments Scott Aaronson.
Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson UC Berkeley IAS.
Quantum Computing and Dynamical Quantum Models ( quant-ph/ ) Scott Aaronson, UC Berkeley QC Seminar May 14, 2002.
Limitations of Quantum Advice and One-Way Communication Scott Aaronson UC Berkeley IAS Useful?
Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley
How Much Information Is In A Quantum State? Scott Aaronson MIT |
Quantum Double Feature Scott Aaronson (MIT) The Learnability of Quantum States Quantum Software Copy-Protection.
An Invitation to Quantum Complexity Theory The Study of What We Cant Do With Computers We Dont Have Scott Aaronson (MIT) QIP08, New Delhi BQP NP- complete.
BQP/qpoly EXP/poly Scott Aaronson UC Berkeley. BQP/qpoly Class of languages recognized by a bounded-error polytime quantum algorithm, with a polysize.
A Full Characterization of Quantum Advice Scott Aaronson Andrew Drucker.
New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson Parts based on joint work with Alex Arkhipov.
Pretty-Good Tomography Scott Aaronson MIT. Theres a problem… To do tomography on an entangled state of n qubits, we need exp(n) measurements Does this.
How to Solve Longstanding Open Problems In Quantum Computing Using Only Fourier Analysis Scott Aaronson (MIT) For those who hate quantum: The open problems.
Scott Aaronson BQP und PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at lastbut.
Scott Aaronson Institut pour l'Étude Avançée Le Principe de la Postselection.
Arthur, Merlin, and Black-Box Groups in Quantum Computing Scott Aaronson (MIT) Or, How Laci Did Quantum Stuff Without Knowing It.
QMA/qpoly PSPACE/poly: De-Merlinizing Quantum Protocols Scott Aaronson University of Waterloo.
Oracles Are Subtle But Not Malicious Scott Aaronson University of Waterloo.
The Equivalence of Sampling and Searching Scott Aaronson MIT.
Scott Aaronson (MIT) BQP and PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at.
New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.
Solving Hard Problems With Light Scott Aaronson (Assoc. Prof., EECS) Joint work with Alex Arkhipov vs.
Scott Aaronson (MIT) Based on joint work with John Watrous (U. Waterloo) BQP PSPACE Quantum Computing With Closed Timelike Curves.
Shortest Vector In A Lattice is NP-Hard to approximate
Approximate List- Decoding and Hardness Amplification Valentine Kabanets (SFU) joint work with Russell Impagliazzo and Ragesh Jaiswal (UCSD)
Foundations of Cryptography Lecture 2: One-way functions are essential for identification. Amplification: from weak to strong one-way function Lecturer:
Space complexity [AB 4]. 2 Input/Work/Output TM Output.
Quantum Information and the PCP Theorem Ran Raz Weizmann Institute.
Quantum Computing MAS 725 Hartmut Klauck NTU
Lecture 16: Relativization Umans Complexity Theory Lecturess.
Scott Aaronson (MIT) Forrelation A problem admitting enormous quantum speedup, which I and others have studied under various names over the years, which.
Complexity 18-1 Complexity Andrei Bulatov Probabilistic Algorithms.
Avraham Ben-Aroya (Tel Aviv University) Oded Regev (Tel Aviv University) Ronald de Wolf (CWI, Amsterdam) A Hypercontractive Inequality for Matrix-Valued.
Complexity ©D. Moshkovitz 1 And Randomized Computations The Polynomial Hierarchy.
Scott Aaronson (MIT) Andris Ambainis (U. of Latvia) Forrelation: A Problem that Optimally Separates Quantum from Classical Computing H H H H H H f |0 
1 Quantum NP Dorit Aharonov & Tomer Naveh Presented by Alex Rapaport.
Space complexity [AB 4]. 2 Input/Work/Output TM Output.
Is Communication Complexity Physical? Samuel Marcovitch Benni Reznik Tel-Aviv University arxiv
One Complexity Theorist’s View of Quantum Computing Lance Fortnow NEC Research Institute.
Quantum Computing MAS 725 Hartmut Klauck NTU
PROBABILISTIC COMPUTATION By Remanth Dabbati. INDEX  Probabilistic Turing Machine  Probabilistic Complexity Classes  Probabilistic Algorithms.
You Did Not Just Read This or did you?. Quantum Computing Dave Bacon Department of Computer Science & Engineering University of Washington Lecture 3:
A limit on nonlocality in any world in which communication complexity is not trivial IFT6195 Alain Tapp.
Umans Complexity Theory Lectures Lecture 1a: Problems and Languages.
1 Introduction to Quantum Information Processing CS 467 / CS 667 Phys 467 / Phys 767 C&O 481 / C&O 681 Richard Cleve DC 3524 Course.
1 Introduction to Quantum Information Processing CS 467 / CS 667 Phys 667 / Phys 767 C&O 481 / C&O 681 Richard Cleve DC 653 Lecture.
1 Introduction to Quantum Information Processing CS 467 / CS 667 Phys 467 / Phys 767 C&O 481 / C&O 681 Richard Cleve DC 3524 Course.
An Introduction to Quantum Computation Sandy Irani Department of Computer Science University of California, Irvine.
1 Introduction to Quantum Information Processing CS 467 / CS 667 Phys 467 / Phys 767 C&O 481 / C&O 681 Richard Cleve DC 3524 Course.
Probabilistic Algorithms
Complexity-Theoretic Foundations of Quantum Supremacy Experiments
Scott Aaronson (MIT) QIP08, New Delhi
3rd Lecture: QMA & The local Hamiltonian problem (CNT’D)
Quantum Computing and the Quest for Quantum Computational Supremacy
Quantum Computation and Information Chap 1 Intro and Overview: p 28-58
Oracle Separation of BQP and PH
Oracle Separation of BQP and PH
Presentation transcript:

Are Quantum States Exponentially Long Vectors? Scott Aaronson (who did and will have an affiliation) (did: IASwill: Waterloo) Distributions over n-bit strings 2 n -bit strings

The Computer Science Picture of Reality Quantum computing challenges this picture Thats why everyone should care about it, whether or not quantum factoring machines are ever built + details

As far as I am concern[ed], the QC model consists of exponentially-long vectors (possible configurations) and some uniform (or simple) operations (computation steps) on such vectors … The key point is that the associated complexity measure postulates that each such operation can be effected at unit cost (or unit time). My main concern is with this postulate. My own intuition is that the cost of such an operation or of maintaining such vectors should be linearly related to the amount of non-degeneracy of these vectors, where the non-degeneracy may vary from a constant to linear in the length of the vector (depending on the vector). Needless to say, I am not suggesting a concrete definition of non-degeneracy, I am merely conjecturing that such exists and that it capture[s] the inherent cost of the computation. Oded Goldreich

(1) Its not easy to explain current experiments (let alone future ones!), if you dont think that quantum states are exponentially long vectors [A. 2004, Multilinear Formulas and Skepticism of Quantum Computing] (2) But its not that bad [A. 2004, Limitations of Quantum Advice and One-Way Communication] My Two-Pronged Response

Prong (1) Quantum states are exponentially long vectors

How Good Is The Evidence for QM? (1)Interference: Stability of e - orbits, double-slit, etc. (2)Entanglement: Bell inequality, GHZ experiments (3)Schrödinger cats: C 60 double-slit experiment, superconductivity, quantum Hall effect, etc. C 60 Arndt et al., Nature 401: (1999)

Exactly what property separates the Sure States we know we can create, from the Shor States that suffice for factoring? DIVIDING LINE

Not precision in amplitudes:Not entanglement across hundreds of particles:Not a combination of the two:

Intuition: Once we accept | and | into our set of possible states, were almost forced to accept | | and | + | as well But we might restrict ourselves to tree states: n-qubit states obtainable from |0 and |1 by a polynomial number of linear combinations and tensor products + |0 1 | |0 1 |1 1 |0 2 |1 2

Definition: The tree size of a pure state | is the minimum number of linear combinations and tensor products in any tree representing | Observation: If then the tree size of | equals the multilinear formula size of f up to a constant factor Question: Can we show that quantum states arising in real (or at least doable) experiments have superpolynomial tree size?

Main Result Even to approximate |C takes tree size n (log n) Can derandomize using Reed-Solomon codes Yields the first function f:{0,1} n with a superpolynomial gap between formula size and multilinear formula size If over the codewords of a binary linear code, then |C has tree size n (log n), with high probability if the generator matrix is chosen uniformly at random from is a uniform superposition

Conjectures States arising in Shors algorithm also have superpolynomial tree size Can show this assuming a number-theoretic conjecture: n (log n) tree size lower bounds can be improved to exponential Can show this under the restriction that linear combinations are manifestly orthogonal

Conjectures (cont) 2D and 3D cluster states (lattices of spins with pairwise interactions) have large (2 (n) ?) tree size Would mean that states with enormous tree sizes have already been seen in the lab, if we believe the condensed- matter physicists (e.g. Ghosh et al., Entangled quantum state of magnetic dipoles, Nature 425:48-51, 2003) Can show that a 1D cluster state of n spins has tree size O(n 4 ) If a quantum computer is in a tree state at every time step, then that computer has a nontrivial classical simulation Can show its simulable in the third level of PH

Prong (2) Its not that bad

Quantum Advice BQP/qpoly: Class of languages decidable by polynomial-size, bounded-error quantum circuits, given a polynomial-size quantum advice state | n that depends only on the input length n Nielsen & Chuang: We know that many systems in Nature prefer to sit in highly entangled states of many systems; might it be possible to exploit this preference to obtain extra computational power?

Maybe BQP/qpoly even contains NP! Obvious Challenge: Prove an oracle separation between BQP/poly and BQP/qpoly Harry Buhrman: Hey Scottwhy not try for an unrelativized separation? After all, if quantum states are like 2 n -bit classical strings, then maybe BQP/qpoly NEEEEE/poly!

Result: BQP/qpoly PP/poly Proof based on new communication result: Given f:{0,1} n {0,1} m {0,1} (partial or total), D 1 (f) = O(m Q 1 (f) logQ 1 (f)) D 1 (f) = deterministic 1-way communication complexity of f Q 1 (f) = bounded-error quantum 1-way complexity Corollary: Cant show BQP/poly BQP/qpoly without also showing PP P/poly Like BPP but with no gap

Alices Classical Message Bob, if you use the maximally mixed state in place of my quantum message, then y 1 is the lexicographically first input for which youll output the wrong answer with probability at least 1/3. But if you condition on succeeding on y 1, then y 2 is the next input for which youll output the wrong answer with probability at least 1/3. But if you condition on succeeding on y 1 and y 2, then y 3 is the … y1y1 y2y2

Technicality: We assume Alices quantum message was boosted, so that the error probability is negligible Claim: Alice only needs to send T=O(Q) inputs y 1,…,y T, where Q = size of her quantum message Proof Sketch: Bob succeeds on y 1,…,y T simultaneously with probability at most (2/3) T. But we can decompose the Q-qubit maximally mixed state as where | 1 is Alices true quantum message. Therefore Bob also succeeds with probability (1/2 Q ).

BQP/qpoly PP/poly Alice is the advisor Bob is the PP algorithm But why can Bobs procedure be implemented in PP? A. 2005: PostBQP = PP, where PostBQP = BQP with postselected measurement outcomes (The PostBQP PP direction is an easy modification of Adleman et al.s proof that BQP PP)

Remarks Contrast with PQP/qpoly = Everything, and Razs recent result that QIP/qpoly = Everything Using similar techniques, I get that for every k, there exists a language in PP that does not have quantum circuits of n k, not even with quantum advice A second limitation of quantum advice (A. 2004): there exists an oracle relative to which NP BQP/qpoly

Conjectures The question of whether classical advice can always replace quantum advice (i.e. BQP/poly = BQP/qpoly) is not independent of ZF set theory Similarly for whether classical proofs can replace quantum proofs (i.e. QCMA = QMA) QMA/qpoly PP/poly Randomized and quantum one-way communication complexities are asymptotically equal for all total Boolean functions f

Summary The state of n particles is an exponentially long vector. Welcome to Quantum World! But for most purposes, its no worse than a probability distribution being an exponentially long vector If youre still not happy, suggest a Sure/Shor separator