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## Quantum Computing

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**Quantum Computing**Intro. & Overview:What’s Quantum Computing All About?**Classical vs. Quantum Computing**• For any digital computer, its set of computational states is some set of mutually distinguishable abstract states. • The specific computational state that is in use at a given time represents the specific digital data currently being processed within the machine. • Classical computing is computing in which: • All of the computational states (at all times) are stable pointer states of the computer hardware. • Quantum computing is computing in which: • The computational state is not always a pointer state.**What is “Quantum Computing?”**• Non-pointer-state computing. • Harnesses these quantum effects on a large, complex scale: • Computational states that are not just pointer states, but also, coherent superpositions of pointer states. • States having non-zero amplitude in many pointer states at the same time! “Quantum parallelism.” • Entanglement (quantum correlations) • Between the states of different subsystems. • Unitary (thus reversible) evolution through time • Interference (reinforcement and cancellation) • Between convergent trajectories in pointer-state space.**Why Quantum Computing?**• It is, apparently, exponentially more time-efficient than any possible classical computing scheme at solving some problems: • Factoring, discrete logarithms, related problems • Simulating quantum physical systems accurately • This application was the original motivation for quantum computing research first suggested by famous physicist Richard Feynman in the early 80’s. • However, this has never been proven yet! • If you want to win a sure-fire Nobel prize… • Find a polynomial-time algorithm for accurately simulating quantum computers on classical ones!**Status of Quantum Computing**• Theoretical & experimental progress is being made, but slowly. • There are many areas where much progress is still needed. • Physical implementations of very small (e.g., 7-bit) quantum computers have been tested and work as predicted. • However, scaling them up is difficult. • There are no known fundamental theoretical barriers to large-scale quantum computing. • Guess: It will be a real technology in ~20 yrs. or so.**Early History**• Quantum computing was largely inspired by reversible computation work from the 1970’s: • Bennett, Fredkin, and Toffoli • Early quantum computation pioneers (1980’s): • Early models not using quantum parallelism to gain performance: • Benioff ‘80, ‘82 - Quantum serial TM models • Feynman ‘86 - Q. models of serial reversible circuits • Margolus ‘86,’90 - Q. models of parallel rev. circuits • Performance gains w. quantum parallelism: • Feynman ‘82 - Suggested faster quantum sims with QC • Deutsch ‘85 - Quantum-parallel Turing machine • Deutsch ‘89 - Quantum logic circuits**More Recent History**• There was a rapid ramp-up of quantum computing research throughout the 1990’s. • Some developments, 1989-present • Refining quantum logic circuit models • What is a minimal set of universal gates for QC? • Algorithms: Shor factoring, Grover search, etc. • Developing quantum complexity theory • What is the ultimate power of quantum computation? • Quantum information theory • Communications, Cryptography, etc. • Error correcting codes, fault tolerance, robust QC • Physical implementations • Numerous few-bit implementations demonstrated**Quantum Logic Networks**• Invented by Deutsch (1989) • Analogous to classical Boolean logic networks • Generalization of Fredkin-Toffoli reversible logic circuits • System is divided into individual bits, or qubits • 2 orthogonal states of each bit are designated as the computationalbasis states, “0” and “1”. • Quantum logic gates: • Local unitary transforms that operate on only a few state bits at a time. • Computation via predetermined seq. of gate applications to selected bits.**Gates without Superposition**• All classical input-consuming reversible gates can be represented as unitary transformations! • E.g., input-consuming NOT gate (inverter) inout0 11 0 in out in out**ABA’B’0 0 0 00 1 0 11 0 1 11 1 1 0**Controlled-NOT • Remember the CNOT (input-consuming XOR) gate? A A’ A A’ B B’ = AB B B’ = AB Example: A B A B**Toffoli Gate (CCNOT)**ABCA’B’C’0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 00 1 1 0 1 11 0 0 1 0 01 0 1 1 0 11 1 0 1 1 01 1 1 1 1 1 A A’=A B B’=B A A’ B B’ C C’ = CAB C C’ (XOR) Now, what happens if the unitary matrix elements are not always 0 or 1?**The Square Root of NOT**• If you put in either basis state (0 or 1) you get a state that appears random when measured… • But if you feed the output back into another N1/2without measuring it, you get the inverse of the original value! • “How is thatpossible?” 0 (50%) 0 (50%) 0 1 N1/2 N1/2 1 (50%) 1 (50%) 0 (50%) 0 1 N1/2 N1/2 1 (50%) 0 (50%) 0 0 N1/2 N1/2 1 (50%)**NOT1/2: Unitary implementation**Prob. ½ Prob. ½**The Hadamard Transform**• A randomizing “square root of identity” gate. • Used frequently in quantum logic networks.**Another NOT1/2**• This one negates the phase of the state if the input state was |0.**Optical Implementation of N1/2**• Beam splitters (semi-silvered mirrors) form superpositions of reflected and transmittedphoton states. “1” “1” “1” “1” “0” “0” “0” laser “1”**Deutsch’s Problem**• Given a black-box function f:{0,1}{0,1}, • Determine whether f(0)=f(1), • But you only have time to call fonce! H H f (N’)1/2**Extended Deutsch’s Problem**• Given black-box f:{0,1}n{0,1}, • and a guarantee that f is either constant or balanced (1 on exactly ½ of inputs) • Which is it? • Minimize number of calls to f. • Classical algorithm, worst-case: • Order 2n time! • What if the first 2n−1 cases examined are all 0? • Function could still be either constant or balanced. • Case number 2n-1+1: if 0, constant; if 1, balanced. • Quantum algorithm is exponentially faster! • (Deutsch & Jozsa, 1992.)**Universal Q-Gates: History**• Deutsch ‘89: • Universal 3-bit Toffoli-like gate. • diVincenzo ‘95: • Adequate set of 2-bit gates. • Barenco ‘95: • Universal 2-bit gate. • Deutsch et al. ‘95 • Almost all 2-bit gates are universal. • Barenco et al. ‘95 • CNOT + set of 1-bit gates is adequate. • Later development of discrete gate sets...**Deutsch: Gen. 3-bit Toffoli gate:**• The following gate is universal: (Where is any irrational number.) a bb a**Barenco’s 2-bit gen. CNOT gate**• where ,,, are relatively irrational • Also works, e.g., for =, =/2: U**Barenco et al. ‘95 results**• Universality of CNOT + 1-bit gates • 2-bit Barenco gate already known universal • 4 1-bit gates + 2 CNOTs suffice to build it • Construction of generalized Toffoli gates • 3-bit version via five 2-bit gates • n-bit version via O(n2) 2-bit gates • No auxilliary bits needed for the above • All operations done “in place” on input qubits. • n-bit version via O(n) 2-bit gates, given 1 work bit**Quantum Complexity Theory**• Early developments: • Deutch’s problem (from earlier): Slight speedup • Deutsch & Jozsa: Exponential speed-up • Important quantum complexity classes: • EQP: Exact Quantum Polynomial - like P. • Polynomial time, deterministic. • ZQP: Zero-error Quantum Polynomial - like ZPP. • Probabilistic, expected polynomial-time, zero errors. • BQP: Bounded-error Quantum Poly. - like BPP. • Probabilistic, bounded probability of errors. • All results relativized, e.g., • O: EQPO (NPO co-NPO) Given a certain black-box, quantumcomputers can solve a certain problemfaster than a classical computer caneven check the answer!**Quantum Algorithms**Part I: Unstructured Search**Unstructured Search Problem**• Given a set S of N elements and a black-box function f:S{0,1}, find an element xS such that f(x)=1, if one exists (or if not, say so). • Any NP problem can be cast as an unstructured search problem. • Not necessarily the optimal approach, however. • Bounds on classical run-time: • (N) expected queries in worst case (0 or 1 sol’ns): • Have to try N/2 elements on average before finding sol’n. • Have to try all N if there is no solution. • If elements are length- bit strings, • Expected #trials is (2) - exponential in . Bad!**Quantum Unstructured Search**• Minimum time to solve unstructured search problem on a quantum computer is: • (N1/2) queries = (2/2) = (21/2) • Still exponential, but with a smaller base. • The minimum # of queries can be achieved using Grover’s algorithm.**Grover’s algorithm:**• 1. Start w. amplitude evenly distributed among the N elements, (xi)=1/N • 2. In each state xi, compute f(xi): • 3. Apply conditional phase shift of if f(xi)=1(Negate sign of solution state.) Uncompute f. x1 xN solutionxs f=0 f=1 x1 xN solutionxs**Grover’s algorithm, cont.**• 4. Invert all amplitudes with respect to the average amplitude: x1 xN solutionxs**Grover’s algorithm, cont.**• 5. Go to step 2, and repeat 0.785 N1/2 times. 1 (xs) # of iterations -1**Shor’s Factoring Algorithm**• Solves the >2000-year-old problem: • Given a large number N, quickly find the prime factorization of N. (At least as old as Euclid!) • No polynomial-time (as a function of n=lg N) classical algorithm for this problem is known. • The best known (as of 1993) was a number field sieve algorithm taking time O(exp(n1/3 log(n2/3))) • However, there is also no proof that an (undis-covered) fast classical algorithm does not exist. • Shor’s quantum algorithm takes time O(n2) • No worse than multiplication of n-bit numbers!**Elements of Shor’s Algorithm**• Uses a standard reduction of factoring to another number-theory problem called the discrete logarithm problem. • The discrete logarithm problem corresponds to finding the period of a certain periodic function defined over the integers. • A general way to find the period of a function is to perform a Fourier transform on the function. • Shor showed how to generalize an earlier algorithm by Simon, to provide a Quantum Fourier Transform that is exponentially faster than classical ones.**Powers of numbers mod N**• Given natural numbers (non-negative integers) N1, x<N, and x, consider the sequence: x0 mod N, x1 mod N, x2 mod N, … = 1, x, x2 mod N, … • If x and N are relatively prime, this sequence is guaranteed not to repeat until it gets back to 1. • Discrete logarithm of y, base x, mod N: • The smallest natural number exponent k (if any) such that xk = y (mod N). • I.e., the integer logarithm of y, base x, in modulo-N arithmetic. Example: dlog7 13 (mod N) = ?**Discrete Log Example**• N=15, x=7, y=13. • x2 = 49 = 4 (mod 15) • x3 = 4·7 = 28 = 13 (mod 15) • x4 = 13·7 = 91 = 1 (mod 15) • So, dlog7 13 = 3 (mod N), • Because 73 = 13 (mod N). 0 1 2 3 4 ×7 ×7 ×7 5 6 7 8 9 12 13 14 10 11 ×7**The order of x mod N**• Problem: Given N>0, and an x<N that is relatively prime to N, what is the smallest value of k>0 such that xk = 1 (mod N)? • This is called the order of x (mod N). • From our previousexample, the orderof 7 mod N is…? 0 1 2 3 4 ×7 ×7 ×7 5 6 7 8 9 12 13 14 10 11 ×7**Order-finding permits Factoring**• A standard reduction of factoring N to finding orders mod N: • 1. Pick a random number x < N. • 2. If gcd(x,N)1, return it (it’s a factor). • 3. Compute the order of x (mod N). • Let r := min k>0: xk mod N = 1 • 4. If gcd(xr/21, N) 1, return it (it’s a factor). • 5. Repeat as needed. • The expected number of repetitions of the loop needed to find a factor with probability > 0.5 is known to be only polynomial in the length of N.**Factoring Example**• For N=15, x=7… • Order of x is r=4. • r/2 = 2. • x2 = 5. • In this case (we are lucky), bothx2+1 and x21 are factors (3 and 5). • Now, how do we compute orders efficiently? 0 1 2 3 4 ×7 ×7 ×7 5 6 7 8 9 12 13 14 10 11 ×7**Quantum Order-Finding**• Uses 2 quantum registers (a,b) • 0 a < q, is the k (exponent) used in order-finding. • 0 b < n, is the y (xk mod n) value • q is the smallest power of 2 greater than N2. • Algorithm: • 1. Initial quantum state is |0,0, i.e., (a=0, b=0). • 2. Go to superposition of all possible values of a:**Quantum Algorithms III**Wrap-up: Classical vs. Quantum Parallelism Quantum Physics Simulations**Classical Unstructured Search**• The classical serial algorithm takes (N) time. • But: Suppose we search in parallel! • Have M<N processors running in parallel. • Each searches a different subset ofN/M elements of the search space. • If processors are ballistic reversible: • Can cluster them in a dense mesh of diameter (M1/3). • Time accounting: • Computation time: (N/M) • Communication time: (M1/3) (at lightspeed) • Total:T N/M + M1/3 is minimized when M N3/4 N1/4 Faster than Grover’s algorithm! M1/3 M**Classical+Quantum Parallelism**• Similar setup to classical parallelism: • M processors, searching N/M items each. • Except, each processor uses Grover’s algorithm. • Time accounting: • Computation: T (N/M)1/2 • Communication: T M1/3 (as before) • Total:T (N/M)1/2 + M1/3 • Total is minimized when MN 3/5 • Minimized total is T N1/5. • I.e., quantum unstructured search is really only N1/4/N1/5 = N1/20 × faster than classical!**Simulating Quantum Physics**• For n particles, takes exponential time using the best known classical methods. • Feynman suggested: use quantum computing • Takes only polynomial time on a QC. • History of some fast QC physics algorithms: • Schrödinger equation (Boghosian, Yepez) • Many-body Fermi systems (Abrams & Lloyd) • Eigenvalue/eigenvector computations (ditto) • Quantum lattice gases, Ising models, etc. • Totally general quantum physics simulations(Field theories, etc.) - Lloyd 1996**Simulating Quantum Computers**on classical ones**Efficient QC Simulations**• Task: Simulate an n-qubit quantum computer. • Maximally stupid approach: • Store a 2n-element vector • Multiply it by a full 2n×2n matrix for each gate op • Some obvious optimizations: • Never store whole matrix (compute dynamically) • Store only nonzero elements of state vector • Especially helpful when qubits are highly correlated • Do only constant work per nonzero vector element • Scatter amplitude from each state to 1 or 2 successors • Drop small-probability-mass sets of states • Linearity of QM implies no chaotic growth of errors**Linear-space quantum simulation**• A popular myth: • “Simulating an n-qubit (or n-particle) quantum system takes e(n) space (as well as time).” • The usual justification: • It takes e(n) numbers even to represent a single (n)-dimensional state vector, in general. • The hole in that argument: • Can simulate the statistical behavior of a quantum system w/o ever storing a state vector! • Result BQP PSPACE known since BV’93... • But practical poly-space sims are rarely described**The Basic Idea**• Inspiration: • Feynman’s path integral formulation of QED. • Gives the amplitude of a given final configuration by accumulating amplitude over all paths from initial to final configurations. • Each path consists of only a single (n)-coordinate configuration at each time, not a full wavefunction over the configuration space. • Can enumerate all paths, while only ever representing one path at a time.**Simulating Quantum Computations**• Given: • Any n-qubit quantum computation, expressed as a sequence of 1-qubit gates and CNOT gates. • An initial state s0 which is just a basis state in the classical bitwise basis, e.g.00000. • Goal: • Generate a final basis state stochastically with the same probability distribution as the quantum computer would do. 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