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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