1R. Rao: Lecture 10 – Quantum Computing
Lecture 10: Quantum Computing
Basic Quantum PhysicsMotivation: Waves versus particles, interference
experimentsQuantum Notation, Jargon, and Definitions
Quantum computingQuantum logic gatesQuantum softwareQuantum hardware
2R. Rao: Lecture 10 – Quantum Computing
Classical double-slit experiment #1 Gun shoots identical particles
Large particles like tennis balls All identical With random direction At a slow firing rate
Probability of detecting particles for two slits is sum of individual slit probabilities: P12 = P1 + P2
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Classical double-slit experiment #2 Source generates water waves
At any intensity value No reflection from absorber Detector measures wave intensity
(wave height)2
Waves interfere at absorber
Wave intensity I12 = |h1 + h2|2 = |h1|2 + |h2|2 + 2|h1||h2|cos h1 and h2 are complex numbers
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Double-slit with electrons Gun shoots electrons
Individual particles Indestructible All identical With random direction At a slow firing rate
Detector sees individual electrons P12 P1 + P2
P12 = |1 + 2|2 (interference!) 1 and 2 are complex numbers Electrons exhibit wave behavior?
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Measurement: Watching the electrons
Electrons scatter light Put a light at the back
side of the slit wall Watch where the
electron goes
Light on: no interference! P12
= P1 + P2
Light off: interference! P12 = |1 + 2|2
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First principles of quantum mechanics
1. The probability of an event is given by the square of the absolute value of the probability amplitude for that event:
P probability probability amplitude (complex)P | |2
2. When an event can occur several ways, the probability amplitude for the event is the sum of the individual probability amplitudes:
1 + 2 (no measurement interference)P |1 + 2 |2 (probability of event e.g. electron at backstop)
3. If you measure and determine which of the possible alternatives an experiment takes, then the probability of the event is the sum of the probabilities for each alternative:P12 = P1 + P2 = |1|2 + |2|2 (measurement at slits no interference)
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Other Examples of Quantum Phenomena
Spin of an electron: spin up or spin down Can be set to a continuum of values but collapses to up or down
when measured with a magnetic field
Polarization of a photon: horizontal or vertical Measured using a calcite crystal; can be set to a continuum of values
but collapses to horizontal or vertical upon measurement
Energy levels of an ion: excited or ground state Can be put in a continuum of states in between these two but
collapses to excited or ground state when measured
Just as the state of a transistor can represent a bit (0 or 1), the state of a quantum system (e.g. spin) represents a qubit
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Classical notation for a bit: x = 0 or x = 1 (only 2 values)
Dirac notation for a quantum bit x (Qubit |x>) e.g. spin of an electron: up = |1>, down = |0>, continuum of possible values |x>
Quantum Notation
01
0
x = 0(1,0)
(0,1)
x = 1
10
1 10 21 ccx Qubit |x>
|1>
|0>Example: c1 and c2 are two real numbers
9R. Rao: Lecture 10 – Quantum Computing
A qubit in a quantum system can exist in a linear superposition of basis states (“eigenstates”) |0> and |1>:
c1and c2 are complex numbers: c1= a1 + b1i ; c2= a2 + b2i i is the square root of –1:
Quantum Jargon
10 21 ccx
)1i(i.e.1i 2
2
1
cc
122
21 cc
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Cliff’s notes on complex numbers
Consider a complex number c = a + bi
Complex conjugate of c is c* = a - bi
Amplitude of c = = amplitude of c*
Squared amplitude of c = a2 – (-b2) = a2 – i2b2 = c*c
c can also be written as c = Aei = A (cos + i sin ) where:
is the amplitude, is the phase of c,
222 bac
sin and cos2222 ba
b
ba
a
22 baA
11R. Rao: Lecture 10 – Quantum Computing
The Effect of Measurement
If you measure the quantum system, the qubit (superposition of states) collapses to one of the basis states |0> and |1>
c1and c2 are called probability amplitudes because probability of getting |0> or |1> upon measurement depends on their squared amplitudes:
Since Prob(|0>) + Prob(|1>) = 1,
1or0either measure10 21 ccx
21
21
210Prob bac 2
222
221Prob bac
122
21 cc
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Unitary matrices
Matrix of complex numbers:
Conjugate transpose of a matrix:
A matrix U is unitary if U*TU = I I is the identity matrix
E.g., 2221
1211
cccc
AcA ij
* E.g., **22
*12
*21
*11*
cccc
AcA Tji
T
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Why are unitary matrices important?
Suppose |x> is a qubit:
|x> has length 1
Suppose |y> = U|x> (i.e. transform |x> using U)
Length of |y> is:
Unitary matrices preserve length! Unitary transformations conserve probability
1)(2
1**2
*1
2
1*2
*1*
cc
UUcccc
Ucc
UxUlength T
T
122
21 cc
2
1
cc
x
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Quantum Operations and Gates
Quantum systems are described by Schrödinger’s wave equation
Integrating this differential equation, we can show that the state |x> of a quantum system evolves as: |x>new = U|x>old where
U is a unitary matrix derived from Hamiltonian HH is a unitary matrix that represents total energy of the system Every quantum state |x> is a vector of unit length Quantum gates map unit vectors |x> to unit vectors |y>
For quantum computing, we design H so that U acts like a logic gateQuantum computers are deterministic, linear, reversible, and
unitary until a measurement is made
tHdtiU
0'exp
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1-bit quantum gates
Example: NOT gate
Check: UNOT is unitary
1-bit gates:
UNOT FHG IKJ0 11 0
U
U
NOT
NOT
10
01
01
10
FHGIKJ FHGIKJFHGIKJ FHGIKJ
S phase shift gate
R Rotation gate
US FHG IKJ1 00 1
U R FHG IKJ1
21 11 1
Walsh-Hadamard gate
11
112
1WHU
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1-bit quantum gates (cont.)
There are infinitely many 1-bit gates, corresponding to the rotations about a sphere: e.g. R gate
221
221
221
221
NOT ii
ii
U
Another example:
Square root of NOT gate
NOTNOTNOT 0110
UUU
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More Notation
A qubit is given by a column vector:
In Dirac notation, define a row vector:
The “bra”-“ket” of x is the inner product:
2
1
cc
x
12*21
*1
2
1*2
*1
cccc
cc
ccxx
lyrespective of ket"" and bra"" called are and xxx
*2
*1 ccx
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Orthonormal basis states
The vectors |x and |y are orthonormal iff they are orthogonal:
and normalized:
Example:
x y 0
x x y y 1 1 and
001
1001 and 010
0110
110
1011 and 101
0100
10
1and01
0
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What is a Hilbert Space? If S is a set of basis states, then the Hilbert space is the space
of functions from S to complex numbers: each function produces a vector of complex numbers in this space
Example: |0> and |1> define an orthonormal Hilbert space whose vectors are of the form:
The states of a quantum system are vectors in a particular Hilbert space
Measurement of the system produces one of the orthogonal axes (a basis state or “eigenstate”) of the Hilbert space
2
1
cc
x
20R. Rao: Lecture 10 – Quantum Computing
Vector Products in Dirac Notation
Vector Products of basis states:
gate! NOT for thematrix theis This 0110
0110
0100
0110
01
0010
1001
10
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Quantum Logic Gates: 1-bit gates
NOT gate:
NOT gate in Dirac Notation:
UNOT FHG IKJ0 11 0
U
U
NOT
NOT
10
01
01
10
FHGIKJ FHGIKJFHGIKJ FHGIKJ 01
10
NOT
NOT
U
U
0110NOT U
01011101
10010100
NOT
NOT
U
U
22R. Rao: Lecture 10 – Quantum Computing
2-bit Quantum Logic Gates Orthonormal Basis States:
Controlled NOT gate:
CN gate in Dirac Notation:
A B A’ B’0 0 0 00 1 0 11 0 1 11 1 1 0
1011111001010000 CNU
11and 10,01,00
1000
11
0100
10
0010
01
0001
00
1111101000100010 E.g. CNU
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Two-bit Quantum Operations
Operation of CN gate on arbitrary qubits x:
2
3
1
0
3
2
1
0
3210
3210
3210
10110100
11100100
,11100100 If
cccc
cccc
U
cccc
UcUcUcUcxU
ccccx
CN
CNCNCNCNCN
1011111001010000 CNU
UCN switches the amplitudes of |10> and |11>
24R. Rao: Lecture 10 – Quantum Computing
Entanglement
Two bits in a quantum system are entangled if measurement of one is always correlated with measurement of the other
If you measure the second bit of x, you know the first bit also: E.g. if second bit was 1, we know first bit must be 1
The qubits in y are not entangled: cannot determine value of first bit
based on value of second bit
Entangled qubits cannot be factored into their components:
zero) are and ( 1100 E.g. 2130 ccccx
zero) are and ( 1101 E.g. 2130 ccccy
10101100 221130 babacc
25R. Rao: Lecture 10 – Quantum Computing
No Cloning Theorem There is no unitary transform that allows us to copy a qubit
Proof: Suppose U is a copying matrix: U|c0> = |cc> for all states |c> Then,
Illustration: Copy a bit using CNOT….yields an entangled state!
)1100(2
1)11100100(21 but
)1100(2
1 ),10(2
1 if
cccU
cUc
10 ba
0
26R. Rao: Lecture 10 – Quantum Computing
Three-bit Quantum Gates
Controlled Controlled NOT (CCN) gate:
UCCN is an 8 x 8 matrix:
UCCN is universal: we can form any Boolean function using only CCN gates: e.g. AND if C = 0
A CCN gate
A A'
B B'
C C'
A B C A’ B’ C’0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 00 1 1 0 1 11 0 0 1 0 01 0 1 1 0 11 1 0 1 1 11 1 1 1 1 0
110111111110000000
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Premise of quantum computing
Simulating an N-bit quantum system on a classical computer requires an amount of computation exponential in N Need to track 2N complex amplitudes simultaneously
Nature updates real systems in constant time Updates all the amplitudes simultaneously Uses quantum superposition, or quantum parallelism
Quantum computation: Use nature to compute Use N qubits to represent 2N complex amplitudes
Perform unitary operations on qubitsMeasure to get the output
Harness quantum superposition to get exponential speedupexponential speedup
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Comparson of Classical versus Quantum Computing
N particles 2N unique states
Computations are sequential
Select 1 state Operate on it Put it back into memory
Example: N=3 3 bits 8 states Work on one number at a time
x = 101
N particles 2N unique states
Computations occur in parallel States interact
They are entangled Operate on all states at once
Example: N=3 3 qbits 8 complex amplitudes Operator manipulates 8 at once
x a b c h
1 000 001 010 111...
Classical Computer Quantum Computer
Exponential SpeedupExponential Speedup
29R. Rao: Lecture 10 – Quantum Computing
Shor’s Quantum Factoring Algorithm
Suppose you want to factor a number N
Shor’s algorithm:1. Pick random x < N.2. Compute f = gcd(x,N); if f l, return f // f is a factor3. Find the least r > 0 such that xr 1 (mod N).4. Compute f1 = gcd(xr/2 – 1,N) ); if f1 l, return f1 // f1 is a factor5. Compute f2 = gcd(xr/2 + 1,N); if f2 l, return f2 // f2 is a factor6. Go to 1 and repeat
Number of repetitions for finding a factor with prob > 0.5 is polynomial in length of N
Hard part: Step 3. Find the least r such that xr 1 (mod N). r is the period of repetition of x1, x2,… (mod N).
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Quantum parallelism for finding period Finding the period r of repetition of x1, x2,… (mod N).
1. Prepare an equal superposition of all values of r < q = N2
2. Chose random x and compute (xr mod N) for all r simultaneously:
3. Apply quantum Fourier Transform UQFT to superposition of states:
4. Measure contents of register containing k to compute period r See Shor’s paper and tutorials on class website for more details
1
1
0,q
r
qr
q
r
r qNxr1
mod,
q
kr
r qNxkqikr0,
mod,)/2exp(
31R. Rao: Lecture 10 – Quantum Computing
Grover’s Database Search Algorithm Problem: Search a random list of N items for a target item
xT such that the function P(xT) is true e.g. searching for a key in DES
Grover’s algorithm: Amplify amplitude of target item
1. Prepare an equal superposition of all x2. Invert the amplitude of xj if P(xj) = 13. Subtract all amplitudes from average amplitude4. Repeat (2) and (3) times5. Measure the result
Quadratic speedup over classical search (O(N) steps).
N4
32R. Rao: Lecture 10 – Quantum Computing
Quantum Hardware Suggested Possibilities: Ion Traps (Quantum dots), Cavity QED
(quantum electrodynamics), NMR: nuclear magnetic resonance
QC Features Ion Trap Cavity QED NMRQubit Energy levels in an
ion within an electric field
Polarization of a photon in a cavity
Spin states of a nucleus in a molecule
Preparation Ion cooling Prepare linearly and circularly polarized photons
Set average state of spins in sample
Evolution Apply laser pulses at specific frequencies
Photon-photon interactions
Apply radio frequency pulses
Conditional logic
Coupled vibrations of trapped ions
State of Cesium ion in cavity & photon polarization
Nuclear spin-spin interactions in a molecule
33R. Rao: Lecture 10 – Quantum Computing
Quantum Computing: Summary
Basic Mechanism: Parallel computation along all possible computational paths, with selective manipulation of probability amplitudes
Main Features: Problem instances encoded as states of a quantum system (e.g. spins of n electrons, polarization values of n photons etc.) 1. The system is put into a superposition of all possible states, each
weighted by its probability amplitude (= a complex number ci)E.g. Qubits for 2 electrons = c1 |00> + c2 |01> + c3 |10> + c4 |11>
2. The system evolves according to quantum principles: 1. Unitary matrix operation: describes how superposition of states
evolves over time when no measurement is made2. Measurement operation: maps current superposition of states to
one state based on probability = square of amplitude ci
E.g. probability of seeing output bits (00) is | c1|2
34R. Rao: Lecture 10 – Quantum Computing
Quantum Computing: Problems and Future Directions
Problems: Decoherence: Environmental noise may inadvertently “measure” the
system, thereby disturbing the computation (current decoherence time ~ 1 ms)Software solution: Error correcting codes may help ([Shor et al.])
Scaling: All physical implementations so far (NMR, Cavity QED, etc.) have failed to scale beyond a few qubits.
Future Directions: Hardware Implementations: New physical substrates are needed that
allow manipulations of large numbers of qubits (superpositions of states) with little or no decoherence
New Algorithms: New ways of exploiting quantum parallelism are needed that allow solutions to NP-complete problems
35R. Rao: Lecture 10 – Quantum Computing
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