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Quantum Computing: Whats It Good For? Scott Aaronson Computer Science Department, UC Berkeley January 10, 2002 www.cs.berkeley.edu/~aaronson

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Overview 1. History and background 2. The quantum computation model 3. Example: Simons algorithm 4. Other algorithms (Shors, Grovers) 5. Limits of quantum computing, including recent work 6.The future

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Richard Feynman (1981):...trying to find a computer simulation of physics, seems to me to be an excellent program to follow out...and I'm not happy with all the analyses that go with just the classical theory, because nature isnt classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem because it doesn't look so easy.

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David Deutsch (1985): Computing machines resembling the universal quantum computer could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine … Complexity theory for [such machines] deserves further investigation.

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What Is Quantum Mechanics?

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Framework for atomic-scale physical theories Computational model with amplitudes instead of probabilities Complicated (lots of integral signs) Simple Pessimistic (i.e. Heisenberg uncertainty relation) Optimistic (i.e. Shors factoring algorithm) Traditional Physics ViewQuantum Computing View What Is Quantum Mechanics?

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The Model Computer has n bits of memory Classical case: if n=2, possible states are 00, 01, 10, 11 Randomized case: States are vectors of 2 n probabilities in [0,1] i.e. Pr[00]=0.2 Pr[01]=0.2 Pr[10]=0.1 Pr[11]=0.5 Quantum case: States are vectors of 2 n complex numbers called amplitudes

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The Model (cont) Dirac ket notation: We write state as, i.e., 0.5 |00 - 0.5 |01 + 0.5i |10 - 0.5i |11 Superposition over basis states Normalization: If state is i i |i, then i | i | 2 = 1 (Why complex numbers? Why | i | 2 and not i 2 ?)

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Measurement When we measure state, see basis state |i with probability | i | 2 Furthermore, state collapses to |i Can also make partial measurements Example: Measuring 1 st bit of yields |00 with ½ prob., (|10 +|11 )/ 2 with ½ prob.

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Time Evolution Matrix U is unitary iff UU =I, conjugate transpose Equivalently: U preserves norm Can multiply amplitude vector by some unitary U (i.e. replace state | by U| ) Quantum analogue of Markov transitions

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Example: Square Root of NOT Hadamard matrix: H |0 = (|0 +|1 )/ 2H|1 = (|0 -|1 )/ 2 H (|0 +|1 )/ 2 = |0 H(|0 -|1 )/ 2 = |1

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Quantum Circuits Unitary operation is local if it applies to only a constant number of bits (qubits) Given a yes/no problem of size n: 1.Apply order n k local unitaries for constant k 2.Measure first bit, return yes iff its 1 BQP: class of problems solvable by such a circuit with error probability at most 1/3 (+ technical requirement: uniformity)

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The Power of Quantum Computing Bernstein-Vazirani 1993: BPP BQP PSPACE BPP: solvable classically with order n k time PSPACE: solvable with order n k memory Apparent power of quantum computing comes from interference -Probabilities always nonnegative -But amplitudes can be negative (or complex), so paths leading to wrong answers can cancel each other out

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Simons Problem Given a black box x f(x) Promise: There exists a secret string s such that f(x)=f(y) y=x s for all x,y ( : bitwise XOR) Problem: Find s with as few queries as possible

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Example Input xOutput f(x) 0004 0012 0103 0111 1002 1014 1101 1113 Secret string s: 101 f(x)=f(x s)

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Simons Algorithm Classically, order 2 n/2 queries needed to find s - Even with randomness Simon (1993) gave quantum algorithm using only order n queries Assumption: given |x, can compute |x |f(x) efficiently

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Simons Algorithm (cont) 1. Prepare uniform superposition 2. Compute f: 3. Measure |f(x), yielding for some x

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Simons Algorithm (cont) 4. Apply to each bit of Result: where

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Simons Algorithm (cont) 5. Measure. Obtain a random y such that 7. Solve for s. Can show solution is unique with high probability. 6. Repeat steps 1-5 order n times. Obtain a linear system over GF 2 :

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Schematic Diagram O b s e r v e f(x) O b s e r v e |0

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Period Finding Given: Function f from {1…2 n } to {1…2 n } Promise: There exists a secret integer r such that f(x)=f(y) r | x-y for all x Problem: Find r with as few queries as possible Classically, order 2 n/3 queries to f needed Inspired by Simon, Shor (1994) gave quantum algorithm using order poly(n) queries

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Example: r=5

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Factoring and Discrete Log Using period-finding, can factor integers in polynomial time (Miller 1976) Also discrete log: given a,b,N, find r such that a r b(mod N) Breaks widely-used public-key cryptosystems: RSA, Diffie-Hellman, ElGamal, elliptic curve systems…

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Grovers Algorithm Unsorted database of n items Goal: Find one marked item Classically, order n queries to database needed Grover 1996: Quantum algorithm using order n queries

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Limits of Quantum Computing Bennett et al. 1996: Grovers algorithm is optimal (Quantum search requires order n queries) Beals et al. 1998: For all total Boolean functions f: {0,1} n {0,1}, if quantum algorithm to evaluate f uses T queries, exists classical algorithm using order T 6 queries.

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Collision Problem Given: a function f: {1,…,n} {1,…,N}, n even Promise: f is either 1-1 (i.e. 3,7,9,2) or 2-1 (5,2,2,5) Problem: Decide which Models graph isomorphism, breaking cryptographic hash functions Classical algorithm needs order n queries to f Brassard et al. 1997: Quantum algorithm using n 1/3 queries

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Collision Lower Bound Can a quantum algorithm do better than n 1/3 ? Previously couldnt even rule out constant number of queries! A 2001: Any quantum algorithm for collision needs order n 1/5 queries Shi 2001: Improved to order n 1/3

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The Future

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When processor components reach atomic scale, Moores Law breaks down - Quantum effects become important whether we want them or not - But huge obstacles to building a practical quantum computer!

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Implementation

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Key technical challenge: prevent decoherence, or unwanted interaction with environment Approaches: NMR, ion trap, quantum dot, Josephson junction, optical… Recent achievement: 15=3 5 (Chuang et al. 2001) Larger computations will require quantum error- correcting codes

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Quantum Computing: Whats It Good For? Potential (benign) applications - Faster combinatorial search - Simulating quantum systems Makes QM accessible to non-physicists Spinoff in quantum optics, chemistry, etc. Surprising connections between physics and CS New insight into mysteries of the quantum

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1 Recap (I) n -qubit quantum state: 2 n -dimensional unit vector Unitary op: 2 n 2 n linear operation U such that U † U = I (where U † denotes the conjugate.

1 Recap (I) n -qubit quantum state: 2 n -dimensional unit vector Unitary op: 2 n 2 n linear operation U such that U † U = I (where U † denotes the conjugate.

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