Introduction into Quantum Computations
Alexei Ashikhmin
Bell Labs
Workshop on Quantum Computing and its Application
March 16, 2017
 Qubits
 Unitary transformations
 Quantum Circuits
 Quantum Measurements
 Quantum Fourier Transform  Phase Estimation
Order Finding  Fast Factoring
 Quantum Error Correction
Dirac vs. Linear Algebra Notation
is the inner product
If
then
Dirac vs. Linear Algebra Notation (cntd)
Vectors
Hence
is an orthonormal basis of
can be written in the form
We can also say
and write
Typically we assume
is an orthonormal basis of
Qubits
Laser beams
Electron can be in two states:
Ground (G) |0> and Excited |1>
– Laser 1: electron moves from G to
E state
– Laser 2: electron moves from E to
G state
– Laser 3: electron moves to a
superposition of states, e.g. 30%
G and 70 % E
Postulate 1: Pure state of a qubit is
In our example
n Qubits
qubits
Postulate 1 The state (pure) of
•
qubits is a vector
hence, manipulating by qubits,
we effectively manipulate by
complex coefficients
• As a result we obtain a significant (sometimes exponential)
speed up
Unitary Evolution
Postulate 2 The time evolution of a closed quantum system is
described by the Schrodinger equation
•
is the system Hamiltonian,
• The solution of this equation is
is unitary operator
Unitary Evolution
Postulate 2’ The time evolution of a closed quantum system is
described by a unitary transformation
Apply a unitary rotation
state
new state
Quantum Circuits
Quant Not Gate
LA notation:
Quantum Circuits
Controlled Not Gate (Quant XOR Gate)
two qubits in the joint state:
This circuit computes the Boolean function
for ALL inputs simultaneously!
the same qubits
(particles) but in a
new state:
Quantum Circuits
• Classical AND and NOT gates form a universal set, i.e., they
allow one to implement any Boolean function
• Hadamard, Phase, CNOT, and
gates form a universal
set, i.e., they allow one to approximate any unitary with
arbitrary precision
• Can we approximate any given
circuit of size polynomial in ?
NO!
unitary
using a
von Neumann Measurement
and
orthogonal subspaces; they span
is the orthogonal projection on
is the orthogonal projection on
Postulate 3
•
is projected on
with probability
•
is projected on
with probability
• We know to which subspace
was projected
quant. output
classical output shows
to which subspace
or
the state
was projected
Quantum Fourier Transform
• Discrete Fourier Transform (DFT) of size N is defined as
• Quantum DFT is defined by
QDFT
or
QDFT
Quantum Fourier Transform
• Let
and
• After some computations one gets
QDFT
• The final state is a tensor product of individual states of n
qubits. Typically this means that it is not difficult to construct a
quantum circuit for it.
Quantum Fourier Transform
• Example. QDFT Circuit for
Quantum Fourier Transform
• The complexity of QDFT is
• The complexity of Classical DFT is
• The complexity of Classical DFT that finds (with possible
error) the largest coefficient of DFT is
• Can we use QDFT to get coefficients
NO!
in
Phase Estimation
• Let
• So
be a unitary operator and
its eigenvector.
• We would like to find the phase
• Phase Estimation has multiple applications
Phase Estimation
Inverse
QDFT
Order Finding
• x and N are coprime. We need to find the smallest r s
Example
No classical algorithms with complexity
Quantum approach. Take
For
s.t.
, we have
Order Finding
•
is s.t.
• For
Theorem
, we have
is unitary with
eigenvalues
and
eigenvectors:
and
bits
Order Finding
We use this
in the phase estimation circuit with input
Inverse
QDFT
At the output we get
for random
Fast Factoring
Finding the Greatest Common Devisor (gcd) of integers z and N
is easy. Complexity
Algorithm
1. Take random
2. Find its order, i.e., the smallest
3. If
a. is even
b.
then find
s.t.
Theorem Either
or (and)
is a nontrivial factor of N
Quantum Errors
• Quantum computer is unavoidably vulnerable to errors
• Any quantum system is not completely isolated from the
environment
• Uncertainty principle – we can not simultaneously reduce:
– laser intensity and phase fluctuations
– magnetic and electric fields fluctuations
– momentum and position of an ion
• The probability of spontaneous emission is always greater
than 0
• Leakage error – electron moves to a third level of energy
Depolarizing Channel (Standard Error Model)
Depolarizing Channel
means the absence of error
are the flip, phase, and flip-phase errors respectively
This is an analog of the classical quaternary symmetric channel
No-Cloning Theorem
• Perhaps the simplest classical error correcting code is
repetition code.
Encoding: 0  00000, 111111
• So if say 2 bits are flipped (01001) we still can say it was 0
Can we use the same idea for quantum error protection? No.
• We have a qubit in unknown state
• We bake our own qubit in any desirable state, say
• The joint state of
is
Theorem (No-Cloning) There is no unitary transform
s.t.
Quantum Codes
1
2
…
k
information qubits
in state
k+1 …
n
unitary rotation
1
2
… n
quantum codeword
in the state
redundant qubits
the joint state:
,
is a linear subspace of
,
is the code rate
is an [[n,k]] quantum code
Classical Linear Codes
• Def. Binary linear [n,k] code is a k-dimensional subspace of
(all summations and multiplications by modulo 2)
Example. [5,2] code with code vectors:
is its generator, k x n, matrix,
its rows are basis vectors
is its parity check, (n-k) x n, matrix
The minimum distance of this code is d=5
We always have
, where
is an inner product
Quantum Stabilizer Codes
•
•
•
is symplectic inner product:
is a linear [2n, n-k] code, with
and
,
is the dual code with
and
. If
then is self-orthogonal (with respect to symplectic product)
• A self-orthogonal
defines a stabilizer [[n,k]] quant. code
My Work on Quantum Codes
Bounds
• Bounds on the tradeoffs between the code rate
and
error correction capabilities
• Bounds on the probability of undetected error
• Bounds on the probability of error in quantum Hybrid ARQ
• Bounds on Entanglement Assisted quantum stabilizer codes
Constructions
• General construction of nonbinary quantum stabilizer codes
• BCH type quantum stabilizer codes
• Asymptotically good quantum codes with small construction
complexity
Quantum Codes Robust to Decoding Errors (DS codes)
• Bounds, Constructions, Performance of Radom DS codes
Bounds on the Minimum Distance of Quantum Codes
New Upper bounds
Knill, Laflamme’s “Singleton” bound
Existence bound
Robust Quantum Syndrome Measurement
1
1
1
2
2
3
4
Msrmnt
of g1
Msrmnt
of g2
3
Msrmnt
of g3
4
Msrmnt
of g4
5
12
Msrmnt
of g4
Msrmnt
of g1
5
s1(1)
s2(1)
s3(1)
5 qubits in the state
,
is a code vector of [[5,1]] code
s4(1)
s1(2)
s4(3)
Robust Quantum Syndrome Measurement
1
1
2
2
3
4
Msrmnt
of g1
Msrmnt
of g2
3
Msrmnt
of g3
4
Msrmnt
of g4
5
Msrmnt
of f1
5
Decoder of classical [12,4] code
5 qubits in the state
,
is a code vector of [[5,1]] code
12
Msrmnt
of f8
Thank You