Quantum Computation Stephen Jordan

Church-Turing Thesis ● Weak Form: Anything we would regard as “computable” can be computed by a Turing machine. ● Strong Form: Anything we would regard as efficiently computable can be computed in polynomial time by a Turing machine.

Models of Computation ● Turing machines – multiple tapes – multiple read/write heads ● Logic Circuits ● Parallel Computation ● All have been shown polynomially equivalent to Turing machines

Thesis Revised? ● “Computers are physical objects and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics.” -David Deutsch

What Quantum Computers Are ● A reasonable model of computation based on currently known physics ● Apparently more powerful than the Turing machine – can do prime factorization in polynomial time ● The first challenge to the strong Church-Turing thesis.

What Quantum Computers Aren't ● Extant ● A challenge to the weak Church-Turing thesis ● Just like classical computers except smaller and faster ● Analog

Relation To Other Models

Quantum Church-Turing Thesis? ● Many models of quantum computation: – quantum turing machines – quantum circuits – adiabatic quantum computation – measurement based quantum computation – nonabelian anyons ● All have equivalent power (BQP) ● One exception: one clean qubit model

State of The Art ● Quantum Cryptography – fundamentally unbreakable – commercialized – Quantum Computers ● many approaches ● still in the laboratory

Earliest Inklings ● At small scales the laws of classical mechanics break down and quantum mechanics takes over. ● Can computers still work when their components reach this scale? ● Yes: any computation can be made reversible with minimal overhead. [1973] ● Quantum computers can do reversible computation. C. Bennett

Advantages? ● “The full description of quantum mechanics for a large system with R particles...has too many variables. It cannot be simulated with a normal computer with a number of elements proportional to R.” [1982] ● An n-bit number can be factored in time on a quantum computer. [1994] R. Feynman P. Shor

More Advantages ● An unstructured database with N items can be searched in time. L. Grover ● Quantum computers can efficiently simulate quantum systems. ● Quantum computers cannot speed up all problems.

Quantum Mechanics ● The state of a system is represented by a normalized complex vector. ● Example: a bit

Dirac Notation

Inner Product

Two Bits

Dynamics!

Example

Measurement

Quantum Computing ● Start with some state encoding your problem. ● Example: factoring 9 = 1001 ● Apply some sequence of unitary time evolutions. ● Measure, and with high probability obtain a desired result, e.g. 3 = 0011

Quantum Computing ● 2 questions about quantum computing 1)How can we build a quantum computer? We'll ignore this. 2)What can we do with them? We'll turn this into a precise question: For a problem of size n, how many computational steps do we need to solve it on a quantum computer?

● Examples – Find the prime factors of an n-digit number. – Find the shortest route visiting n cities. – Compute for given f. ● Which problems can be solved with fewer steps on quantum computers than on classical computers for large n? Computational Problems

Model of Computation: Quantum Circuits ● Use only k-body interactions, “gates” ● k=2 suffices ● CNOT + one qubit gates suffice ● only finite precision required

Family of Quantum Circuits ● One quantum circuit for each input size ● Trivial Example: bitwise XOR

Circuit Complexity ● Return to our original question: For a problem of size n, how many computational steps do we need to solve it on a quantum computer? ● We can now make it precise: What is the minimum number of gates needed, as a function of n, in a family of quantum circuits which solves the problem?

Problems with Circuit Complexity ● Circuit complexity is notoriously difficult to evaluate ● Explicit circuit families (algorithms) provide upper bounds ● Lower bounds are very difficult, even classically (e.g. P vs. NP)

Query Complexity ● Many problems are naturally formulated in in terms of a blackbox f – Find – Find x s.t. f(x)=1 – Find x which minimizes f ● Classical blackboxes can be made reversible, hence unitary

An Easier Question For a given problem, how many black box queries do we need to solve it on a quantum computer, as a function of problem size? ● Algorithms provide upper bounds. ● Information arguments provide lower bounds. ● Quantum speedups for several black box problems are known. ● In many cases matching quantum lower bounds are known.

Bernstein-Vazirani Problem

Classical Algorithm

Phase Kickback

Bernstein-Vazirani Algorithm

Classical Gradient Estimation ● Classically, you need at least d+1 queries ● Otherwise the system is underdetermined ● Quantumly, one query suffices

Transforms ● Hadamard transform on n bits uses n Hadamard gates ● Quantum Fourier Transform on n bits can be done using gates ● The transforms are on amplitudes! ● Inverse transforms are easy. Just take the adjoint.

Minimizing a Quadratic Form

Further Reading ● Michael Nielsen and Isaac Chuang, Quantum Computation and Quantum Information (2000)

An Optical Analogy

Lower Bounds

Lower Bounds by Polynomials

● After q queries, the amplitudes are polynomials of degree at most q, hence the p(1) is of degree 2q ● Recall that desired result is some boolean function of the blackbox values ● There is a minimal degree for a polynomial to match this function

Paturi's Theorem

Specific Lower Bounds

Other Techniques ● Quantum adversary methods ● Reductions

Further Reading ● E. Bernstein and U. Vazirani, “Quantum complexity theory,” proceedings of STOC 1993 ● S. Jordan, “Fast quantum algorithm for numerical gradient estimation,” Phys. Rev. Lett. 95, (2005) [quant-ph/ ] ● R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. De Wolf. “Quantum lower bounds by polynomials,” Journal of the ACM, Vol. 48, No. 4 (2001) [quant-ph/ ]