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 QuestionFor 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

An Optical Analogy

Lower Bounds

Lower Bounds by Polynomials

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, 050501 (2005) [quant-ph/0405146]

● 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/9802049]