Quantum Computation for
DummiesUW students
Dan Simon
Microsoft Research
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
Enter Quantum Mechanics…
“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)
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 A  BPP 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 n time
Classical Probabilistic Coin flips
H
H
T
1/2
1/2
H
T
H
T
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
T
1/ 2
1/ 2
H
T
H
T
1/2
1/2
1/2
-1/2
=1
=0
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)
1 / 2 1 / 2
1 / 2 1 / 2


• Unitary:
– Squared magnitudes in rows,
columns sum to 1 (2-norm)
– Inverse = Conjugate Transpose
(also unitary)
1 / 2

1 / 2
1/ 2 

1/ 2 
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
H
T
Tag
1/ 2
H
1/2
Tag
Tag
1/ 2
T
1/2
Tag
H
1/2
Tag
T
-1/2
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
H
T
Tag
1/ 2
H
1/2
Tag
Tag
1/ 2
T
1/2
Tag
H
1/2
Tag
T
-1/2
Tag
A Slightly Less Simple Trick
0
0
0
e
Tag
Tag
...
2i [...]
...
n-1
e
Tag
...
Tag
2i [...]
n-1
...
0
...
e
Tag
Tag
n-1
...
2i [...]
e
...
Tag
2i [...]
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