Quantum Computation for Dummies

Dan Simon

Microsoft Research

UW students

The Strong Church-Turing Thesis

• Church-Turing Thesis: Any physically realizable computing machine can be modeled by a Turing Machine (TM)– A statement about the physical world

• Strong Church-Turing Thesis: Any physically realizable computing machine can be modeled by a polynomial-time probabilistic TM (PPTM)– A physical/economic statement of sorts

Consequences of the Thesis

• Some problems just cannot be efficiently solved by real, physical computing machines

• Suspected example: NP-complete problems – NP: Class of problems with polynomial-time

checkable solutions– NP-complete problems: If these are efficiently

solvable, then all NP problems are• Many practical examples, esp. in optimization; e.g., TSP

Challenges to the Thesis

• Moore’s Law: Fageddaboudit– It’s just a matter of time….

• Parallelism: Only a polynomial factor– Like speed, it eventually hits a wall

• Analog: Precision is the catch– Precision is (eventually) as costly as speed

• Chaos: Ditto

“You have nothing to do but mention the quantum theory, and people will take your voice for the voice of science, and believe anything.”

--George Bernard Shaw, Geneva (1938)

Enter Quantum Mechanics…

History

• Benioff (1981): Quantum systems can simulate TM

• Feynman (1982): Can they do more? It appears possible....

• Deutsch (1985): Formalized Quantum TM (QTM) model, constructed an (inefficient) universal QTM (UQTM)

BQP BPPA A

More History

• Deutsch & Jozsa (1992): exponential oracle separation of P (deterministic only) and QP– “promise problem” oracle

• Bernstein & Vazirani, Yao (1993): – efficient UQTM– Equivalence of quantum circuits and QTMs – Superpolynomial oracle separation of BPP

(probabilistic P) and BQP

The Breakthroughs

• Shor (1994): integer factoring, discrete log in BQP

• Grover (1995): General Search in timen

Classical Probabilistic Coin flips

H

H

H

T

T T

H

1/2 1/2

1/4 1/4 1/4 1/4

Probability vs. Amplitude

• Classical probability is a 1-norm– The probability of an event is just the sum of the

probabilities of the paths leading to it

– All the probabilities (for all events) must sum to 1

• In the quantum world, it becomes a 2-norm– Each path has an amplitude

– The amplitude of an event is the sum of the amplitudes of the paths leading to it

– Probability = |Amplitude|2 (for each event)

– All the probabilities (for all events) must (still) sum to 1

Interference

• Amplitudes can be negative (even complex!) and still preserve positive probability

• Different paths can thus “cancel” (negatively interfere with) or “reinforce” (positively interfere with) each other

• Paths are therefore no longer independent– we must consider the entire parallel collection

(superposition) of paths at any given point

Quantum Coin Flips

H

H

H

T

T T

H

2/1 2/1

1/2 1/2 1/2 -1/2

= 0= 1

Another Consequence of Amplitude

• Probabilistic processes (e.g., computation) can be represented by Markov chains (stochastic matrices--to preserve 1-norm)

• Quantum processes are represented by unitary matrices (M-1 = M*) to preserve 2-norm

• Unitary matrices have unitary inverses– hence quantum processes are always reversible– fortunately, that doesn’t exclude classical computing

Stochastic vs. Unitary

• Stochastic:– Rows, columns, sum to 1

(1-norm)

• Unitary:– Squared magnitudes in rows,

columns sum to 1 (2-norm)

– Inverse = Conjugate Transpose (also unitary)

2/12/1

2/12/1

2/12/1

2/12/1

Reversible Computation

• A function is reversibly computable if each step can be computed from the one before it or from the one after it

• Any computable function can be made reversibly computable (at a constant factor cost) if the input is preserved (i.e., the output on input x is (x,f(x)))– Use reversible gates (e.g., Toffoli gates)

– Preserve “work” at each step, then recompute to “clean up”

Exploiting Quantum Effects

• Idea: when searching for needle in haystack…

• ...Just follow all paths by flipping quantum coins, and make the dead ends disappear with negative interference!

• The catch: you must preserve unitarity…– e.g., use Toffoli gates for all your classical

computation, to make it reversible– ….but what else can you do?

A Simple Trick

H T

H

Tag Tag

HH T T2/1 2/1

1/2 1/2 1/2 -1/2

Tag Tag Tag Tag

Coherence

• An “event” can specify the states of multiple objects (coin + tag, multiple coins)

• Multiple paths interfere only if they lead to exactly the same event

• Objects must stay “coherent” for this to work– Superposition must be maintained– In particular, observation destroys coherence– That still permits, e.g., (reversible) computation

A Simple Trick (2)

H T

H

Tag Tag

2/1 2/1

HH T T1/2 1/2 1/2 -1/2

Tag Tag Tag Tag

A Slightly Less Simple Trick

0

0 ...... ... n-1Tag Tag

0 ... n-1 ... 0 ... n-1Tag Tag Tag Tag

Tag ...

[...]2 ie [...]2 ie [...]2 ie [...]2 ie

Shor’s Algorithm for Dummies

• Events with the same tag interfere negatively (i.e., cancel) unless their value “complements” the periodicity of the tags

• Seeing such “complementing” event values reveals the tags’ (possibly unknown) period…

• …Which corresponds to the order of an element in the multiplicative group mod n

• That’s enough information to factor n

Limitations

• The Church-Turing thesis is unaffected (QM is computable--in PSPACE, even)

• Some indication that NP may not be in BQP– Algorithm would have to be “non-relativizing”

• Known methods haven’t (yet) extended to some natural, ostensibly similar problems– Graph isomorphism– Lattice problems

Obstacles

• Getting those funny amplitudes just right – Precision on the quantum scale is required

• Keeping them just right – Error correcting codes needed ([Shor et al.])

• Preventing decoherence– Manipulation and coherence are at cross-purposes– Computing mechanisms themselves may

encourage decoherence

Implementation?

• Various proposals – particle spins, energy states to represent bits

• Best so far: NMR-based implementation of Grover’s search on 4-item “database”– Unlikely to scale well

• Unknown if any implementation can scale well– Practical limits of coherence are still a mystery