# Quantum Computing for Computer Scientists

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Quantum Computing
for
Computer Scientists

Research by Forrest Briggs
Aknowledgements

Townsend, Chris Stone, Peter Saeta

Quantum Computation and Quantum
Information by Nielsen and Chuang
Why should you care about QC?
• It’s inevitable as transistors shrink
• There are quantum algorithms that are faster than any
known classical algorithm
• It sheds light on fundamental issues in theoretical
computer science
• Quantum information processing is here now
• There are abstract parts of quantum computing that will
not depend on how they are physically realized in the
future
• We can simulate quantum computers now
• It’s time to start designing algorithms and gaining insight
from theoretical inquiry
The Qubit
c0 
| ψ >= c0 | 0 > +c1 |1 >                   | ψ >=  
 c1 

€   |Ψ> is the state of the system. It is a superposition of the basis
€
vectors |0> and |1> with complex coefficients.
A Qubit in Nature

Before encountering the magnet, the atom is in the state
1        1
| ψ >=    |↑> +    |↓>
2        2
Multiple Qubits
Two Qubits
| ψ >= c00 | 00 > +c01 | 01 > +c10 |10 > +c11 |11 >

N Qubits
| ψ >= c0 | 0 > +c1 |1 > +... + c2 N −1 | 2 N −1 >

€
Unitary Operators

Every transformation you apply to the state of a
quantum computer is a unitary operator, from a
single gate to an entire algorithm.

An operator A is unitary iff A† = A-1.

Unitary operators are by definition, invertible.
Unitary Operators

A unitary operator on a single qubit is just a 2x2 matrix.

a0                    w       xa0  b0 
| ψ1 >=              | ψ 2 >=             =  
 a1                   y       z  a1   b1 

€            €
Unitary Operators
A unitary operator on 3 qubits that acts as a universal logic
gate.
1   0 0 0 0 0 0 0
                  
0   1 0 0 0 0 0 0
0   0 1 0 0 0 0 0
                  
0   0 0 1 0 0 0 0
0   0 0 0 1 0 0 0
                  
0   0 0 0 0 1 0 0
                  
0   0 0 0 0 0 0 1
                  
0   0 0 0 0 0 1 0

€
Measurement is Probabilistic
Qubit Example:
1       1
| ψ >=    |0>+    |1 >
2       2
If the state of a system is
|Ψ> = c0|0> + c1|1> + … + cn|n>,
€   then after measuring it, |Ψ> = |i> with
probability |ci|2.

Other kinds of measurements are possible.
Take a moment to reflect on the
weirdness:
Basically the same rules to we
have seen apply to any physical
system.

Completely isolated systems in the
universe are deterministic, but any
measurement is probabilistic.
Quantum Circuits
1 1 1 
H=        
2 1 −1

1   0 0 0
€
          
0   1 0 0
UCN =           
0   0 0 1
0   0 1 0


€
Classical Universality
Unitary operators are invertible, so quantum
computation must be reversible (no NAND).

We can construct a quantum circuit for any classically
computable function.

A quantum computer can be simulated on a classical
computer (although not efficiently).

The set of computable functions is the same.
Quantum Universality
We can approximate any unitary operator on a single
qubit as a product of a constant number of gates from
a universal set.

We can approximate unitary operators on many
qubits, but not always efficiently.

It is not possible to prepare arbitrary quantum states
efficiently.
Complexity Theory
BPP - The class of decision problems that can be solved with
bounded probability of error in polynomial time on a TM.

BQP - The class of decision problems that can be solved with a
bounded probability of error with a polynomial sized quantum
circuit (which can be generated by a TM).

P  BPPBQP PSPACE EXP

Factoring is in NP, BQP, probably not in BPP.
Quantum Fourier Transform
T
Input:     | ψ >= [c0       c2    ... c N ]
T
Output:    | ψ '>= [ f 0     f2   ...   fN ]
€                  1 N 2 πijk/N
fj =    ∑ e
N k=0
€
Runtime: O(log n)          FFT: O(n log n)

Not as good as you might think: we can’t
€
prepare arbitrary input states.
QFT Circuit

1     0 
Rk =     2 πi/2 k 
0 e          

€
Factoring

Shor’s algorithm is O(n2 log(n) log(log n))
The best known classical algorithm is O(exp[n1/3 log2/3 n])

Shor’s algorithm reduces to order finding (solve xr = 1 mod z).
Order finding reduces to phase estimation (find an eigenvalue).
Phase estimation reduces to the quantum fourier transform.
Quantum Search Algorithm

Given:
f:{0,…,N}{0,1}
f(x) = 1 if x = a, 0 otherwise

We can find a in O( N ) time.

€
That’s just the tip of the iceberg…
• Quantum information (encryption,
compression, teleportation)
• Physical realizations
• Error correcting codes
• Quantum programming languages
• Applications (better AI, simulation of
quantum mechanical systems, etc).

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Description: Quantum computer is a kind of obey quantum mechanics for high-speed mathematical and logical operations, storage and processing of Quantum Information Physics device. When a device processing and calculation of quantum information, quantum algorithm is running, it is a quantum computer. Quantum computer concept stems from the reversible computer studies. The study of reversible computing aims to solve the problem of energy consumption in the computer.