INTRODUCTION TO QUANTUM COMPUTATION
Sean Bae
Quantum
Mechanics &
Computation
Part 1
- Motivation
- Fun video
- History
- Literature
- Applications
Part 2
- Formalism
- Insights
- Algorithms
- State-of-the-art
1 + 1 = ?
i\hbar\frac{\partial|\psi (t)\rangle}{\partial t} = [-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V]|\psi (t) \rangle
|\psi (t)\rangle = ?
P = NP?
Polynomial Time
"Easy problems"
T(n) \in \mathcal{O}(poly(n))
10^{10} + 10^{10} = 2 \times 10^{10}
10^{10} \times 10^{10} = 10^{20}
15 = 3 \times 5
1539218309217839021739021 = ?
3\times 17\times 30180751161134098465471
Non-deterministic Polynomial Time
"hard problems"
T(n) \in \mathcal{O}(2^{n})
P \subseteq NP
Does being able to quickly RECOGNIZE (NP) correct answers mean there's also a quick way to FIND (P) them?
Moore's Law
Extended Church-Turing Thesis
Any "reasonable" model of computation can be simulated on a (probabilistic) Turing Machine with at most polynomial simulation overhead
Axioms of Quantum Mechanics
- Superposition
- Measurement
- Unitary Evolution
|\text{cat}\rangle = \frac{1}{\sqrt{2}}(|\text{dead}\rangle + |\text{alive}\rangle)
- Trap a cat in a box
- A radioactive material in the box will decay with 50% probability and kill the cat in 1 hour
- Wait for 1 hour
- Is the cat is both dead and alive??
Schrödinger's Cat
Bits
0
1
Qubits = Quantum Bits
|0\rangle
|1\rangle
Quantum Parallelism
Quantum Entanglement
100,000 light years
Milky Way
COLLEGE PARK, MD
Milky Way
ME
YOU
ME
YOU
ME
YOU
ME
YOU
|\downarrow\rangle
|\uparrow\rangle
ME
YOU
|\downarrow\rangle
|\uparrow\rangle
ME
YOU
Information travelled faster than the speed of light?
"Spukhafte Fernwirkung!"
("Spooky action at a distance")
Albert Einstein
"God does not play dice with the universe"
Albert Einstein
John Bell
1964
"Don't tell God what to do with his dice"
Niels Bohr
Practical use of entanglement?
ME
YOU
|\psi\rangle
\frac{1}{\sqrt{2}}|\downarrow\uparrow\rangle +\frac{1}{\sqrt{2}}|\uparrow\downarrow\rangle
ME
YOU
|\psi\rangle
Average Human
\sim 10^{27}
Qubits
PARTICLE CHAMBER
College Park
Paris
SCANNER
PARTICLE CHAMBER
SCANNER
PARTICLE CHAMBER
PARTICLE CHAMBER
SCANNER
SCANNER
College Park
Paris
SCANNER
PARTICLE CHAMBER
PARTICLE CHAMBER
SCANNER
College Park
Paris
SCANNER
PARTICLE CHAMBER
PARTICLE CHAMBER
SCANNER
College Park
Paris
SCANNER
PARTICLE CHAMBER
PARTICLE CHAMBER
SCANNER
Paris
College Park
SCANNER
PARTICLE CHAMBER
PARTICLE CHAMBER
SCANNER
Paris
College Park
We are ready to solve some interesting problems.
Computational Complexity Zoo
EXP
PSPACE
NP
NP-Complete
P
Computational Complexity Zoo
EXP
PSPACE
NP
NP-Complete
P
BQP
Bounded Error
Quantum
Polynomial Time
"solves with probability > 2/3"
Peter Shor
1994
Bell Labs
Non-trivial Square Root of 1 modulo N
x^2 \equiv 1\text{ }(\text{mod}\ N) \text{ and }\ x \neq \pm 1
Non-trivial Square Root of 1 modulo N
x^2-1 \equiv 0\text{ }(\text{mod}\ N)
N|(x + 1)(x - 1)
\text{gcd}(N, x + 1) \text{ and gcd}(N, x -1) \text{ are factors of }N
Factoring 15
4^2=16\equiv 1 \text{ }(\text{mod } 15) \text{ and } 4 \neq \pm 1\text{ }(\text{mod } 15)
\text{gcd}(15, 5) = 5 \text{ and gcd}(15, 3) = 3
15 = 3 \times 5
RSA Cryptography
Fun Facts
- Largest number factored is 56,153 [2014]
- Factoring 2048 bit integer (RSA)
- Classical
- 10 years
- $10^6 trillion
- 10^6 terawatts
- Quantum
- 16 hours
- $100 billion
- 10 megawatts
- Classical
Computational Complexity Zoo
EXP
PSPACE
NP
NP-Complete
P
BQP
NP-Complete Problems
- Protein Folding
- Traveling Salesman Problem (TSP)
- Boolean Satisfiability Problem
NP-Complete Problems
- Protein Folding
- Traveling Salesman Problem (TSP)
- Boolean Satisfiability Problem
- Battleship
- Tetris
- Super Mario Bros
- Candy Crush
- Sudoku
5 Minute Break
QM Formalism
|\psi\rangle\in\mathcal{H}
\| |\psi\rangle \| = 1
i\hbar\frac{d}{dt} |\psi (t)\rangle = H(t)|\psi (t)\rangle
\mathcal{H}_{1}\otimes\mathcal{H}_{2}
|\psi\rangle = \sum_{i,j} c_{i,j}|\phi_i\rangle\otimes |\phi_j\rangle
State Space
Non-relativisitc Temporal Evolution
System Composition
QM Formalism
|\psi\rangle\in\mathcal{H}
\| |\psi\rangle \| = 1
i\hbar\frac{d}{dt} |\psi (t)\rangle = H(t)|\psi (t)\rangle
\mathcal{H}_{1}\otimes\mathcal{H}_{2}
|\psi\rangle = \sum_{i,j} c_{i,j}|\phi_i\rangle\otimes |\phi_j\rangle
State Space
Non-relativisitc Temporal Evolution
System Composition
Hilbert Space
QM Formalism
|\psi\rangle\in\mathcal{H}
\| |\psi\rangle \| = 1
i\hbar\frac{d}{dt} |\psi (t)\rangle = H(t)|\psi (t)\rangle
\mathcal{H}_{1}\otimes\mathcal{H}_{2}
|\psi\rangle = \sum_{i,j} c_{i,j}|\phi_i\rangle\otimes |\phi_j\rangle
State Space
Non-relativisitc Temporal Evolution
System Composition
Normalization
QM Formalism
|\psi\rangle\in\mathcal{H}
\| |\psi\rangle \| = 1
i\hbar\frac{d}{dt} |\psi (t)\rangle = H(t)|\psi (t)\rangle
\mathcal{H}_{1}\otimes\mathcal{H}_{2}
|\psi\rangle = \sum_{i,j} c_{i,j}|\phi_i\rangle\otimes |\phi_j\rangle
Non-relativisitc Temporal Evolution
System Composition
Hamiltonian
State Space
QM Formalism
|\psi\rangle\in\mathcal{H}
\| |\psi\rangle \| = 1
i\hbar\frac{d}{dt} |\psi (t)\rangle = H(t)|\psi (t)\rangle
\mathcal{H}_{1}\otimes\mathcal{H}_{2}
|\psi\rangle = \sum_{i,j} c_{i,j}|\phi_i\rangle\otimes |\phi_j\rangle
Non-relativisitc Temporal Evolution
System Composition
Linear theory
\text{if }|\psi_1\rangle\text{ and }|\psi_2\rangle\text{ are solutions, then }
|\psi_1\rangle + |\psi_2\rangle \text{ is also a valid solution: Superposition}
State Space
QM Formalism
|\psi\rangle\in\mathcal{H}
\| |\psi\rangle \| = 1
i\hbar\frac{d}{dt} |\psi (t)\rangle = H(t)|\psi (t)\rangle
\mathcal{H}_{1}\otimes\mathcal{H}_{2}
|\psi\rangle = \sum_{i,j} c_{i,j}|\phi_i\rangle\otimes |\phi_j\rangle
Non-relativisitc Temporal Evolution
System Composition
Tensor Product
State Space
QM Formalism
|\psi\rangle\in\mathcal{H}
\| |\psi\rangle \| = 1
i\hbar\frac{d}{dt} |\psi (t)\rangle = H(t)|\psi (t)\rangle
\mathcal{H}_{1}\otimes\mathcal{H}_{2}
|\psi\rangle = \sum_{i,j} c_{i,j}|\phi_i\rangle\otimes |\phi_j\rangle
Non-relativisitc Temporal Evolution
System Composition
Separable State
State Space
QM Formalism
|\psi\rangle\in\mathcal{H}
\| |\psi\rangle \| = 1
i\hbar\frac{d}{dt} |\psi (t)\rangle = H(t)|\psi (t)\rangle
\mathcal{H}_{1}\otimes\mathcal{H}_{2}
|\psi\rangle = \sum_{i,j} c_{i,j}|\phi_i\rangle\otimes |\phi_j\rangle
Non-relativisitc Temporal Evolution
System Composition
Entangled State
State Space
QM Formalism
\text{Observable } \mathcal{A}\text{ is a self-adjoint operator on }\mathcal{H}
Eigenbasis
Born Rule
Measurement
\mathcal{A} |a_i\rangle = \lambda_i |a_i\rangle
|\psi\rangle \text{ can be expressed as }|\psi\rangle = \sum_i c_i|a_i\rangle
\text{Probability of measurement outcome is } |c_i|^2
\text{New state is }|a_i\rangle
QM Formalism
\text{Observable } \mathcal{A}\text{ is a self-adjoint operator on }\mathcal{H}
Eigenbasis
Born Rule
Measurement
\mathcal{A} |a_i\rangle = \lambda_i |a_i\rangle
|\psi\rangle \text{ can be expressed as }|\psi\rangle = \sum_i c_i|a_i\rangle
\text{Probability of measurement outcome is } |c_i|^2
\text{New state is }|a_i\rangle
Spectral decomposition
QM Formalism
\text{Observable } \mathcal{A}\text{ is a self-adjoint operator on }\mathcal{H}
Eigenbasis
Born Rule
Measurement
\mathcal{A} |a_i\rangle = \lambda_i |a_i\rangle
|\psi\rangle \text{ can be expressed as }|\psi\rangle = \sum_i c_i|a_i\rangle
\text{Probability of measurement outcome is } |c_i|^2
\text{New state is }|a_i\rangle
Complete orthonormal basis
QM Formalism
\text{Observable } \mathcal{A}\text{ is a self-adjoint operator on }\mathcal{H}
Eigenbasis
Born Rule
Measurement
\mathcal{A} |a_i\rangle = \lambda_i |a_i\rangle
|\psi\rangle \text{ can be expressed as }|\psi\rangle = \sum_i c_i|a_i\rangle
\text{Probability of measurement outcome is } |c_i|^2
\text{New state is }|a_i\rangle
Measurement
\text{Systems do not have definite properties until measured}
von Neumann
Many worlds
Copenhagen
(\sum_i c_i|a_i\rangle)\otimes|\text{measurement device ready}\rangle\otimes |\text{observer ready}\rangle
\text{No collapse of wave function}
(\sum_i c_i|a_i\rangle)\otimes|\text{measurement device shows i}\rangle\otimes |\text{observer sees i}\rangle
\text{but does not define what constitutes a measurement}
Many Worlds Interpretation
i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V\Psi
Schrödinger Equation in 1D
i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V\Psi
Assumption:
\Psi \text{ is real}
Complex
i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V\Psi
Assumption:
\Psi \text{ is real}
Complex
Real
i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V\Psi
Assumption:
\Psi \text{ is real}
Complex
Real
Contradiction!
Dimension of Wave Function
\text{Recall in functional space, }\langle\psi |\psi\rangle = \int_{-\infty}^{+\infty} \psi^* \psi dx = 1
\text{Dimension of }[\psi] = [\frac{1}{\sqrt{L}}]
How can we mathematically manipulate qubits to do interesting computation?
Logic Gates
Logic Circuit
\text{Any Turing computable function can be converted into the form:}
f : \{0,1\}^n \rightarrow \{0,1\}^m
Logic Circuit
\text{Any Turing computable function can be converted into the form:}
\text{NAND and FANOUT are universal}
Quantum Gates
|0100\rangle
|\text{result}\rangle
Gate
|0\rangle\otimes |1\rangle\otimes |0\rangle = |0\rangle|1\rangle|0\rangle = |010\rangle
Simplified Notation
Quantum Gates
|0100\rangle
|\text{result}\rangle
Gate
|0\rangle\otimes |1\rangle\otimes |0\rangle = |0\rangle|1\rangle|0\rangle = |010\rangle
Simplified Notation
Evolution operator U act for finite time according to the Schrödinger Equation
Quantum Gates
|000\rangle + |001\rangle +
|\text{result}\rangle
Gate
|0\rangle\otimes |1\rangle\otimes |0\rangle = |0\rangle|1\rangle|0\rangle = |010\rangle
Simplified Notation
One gate operation acts on all components
|010\rangle + |100\rangle + |110\rangle
Quantum Gates
|\psi\rangle
|\text{result}\rangle
Gate
|0\rangle\otimes |1\rangle\otimes |0\rangle = |0\rangle|1\rangle|0\rangle = |010\rangle
Simplified Notation
\text{Can act on }2^{N} \text{ basis states in 1 clock cycle}
Qubit
\mathcal{H} \cong \mathbb{C}^2
|\psi\rangle = \alpha |0\rangle + \beta |1\rangle
|\alpha |^2 + |\beta |^2 = 1
State Space
Generalized Qubit
Normalization
Quantum Gates
X = \text{X gate (NOT)}
H = \text{Hadamard gate}
Z = \text{Z-gate}
X|0\rangle = |1\rangle
X|1\rangle = |0\rangle
Z|0\rangle = |0\rangle
Z|1\rangle = -|1\rangle
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
Quantum Teleportation Protocol
|\Psi\rangle = |\psi\rangle_t\otimes|\psi^{-}\rangle_{a,b}
|\psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)
|\phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)
Quantum Teleportation Protocol
|\Psi\rangle = |\psi\rangle_t\otimes|\psi^{-}\rangle_{a,b}
=\frac{1}{\sqrt{2}}(\alpha |0\rangle_{t0} + \beta |1\rangle_{t1})\otimes(|01\rangle_{a,b} - |10\rangle_{a,b})
|\psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)
|\phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)
\text{Expand the terms}
Quantum Teleportation Protocol
|\Psi\rangle = |\psi\rangle_t\otimes|\psi^{-}\rangle_{a,b}
=\frac{1}{\sqrt{2}}(\alpha |0\rangle_{t0} + \beta |1\rangle_{t1})\otimes(|01\rangle_{a,b} - |10\rangle_{a,b})
=\frac{1}{2}[(\alpha |1\rangle_{b1} - \beta |0\rangle_{b0} ) |\phi^+\rangle_{t,a} + (\alpha |1\rangle_{b1} + \beta |0\rangle_{b0} )|\phi^-\rangle_{t,a}
+ (\beta |1\rangle_{b1} - \alpha |0\rangle_{b0} )|\psi^+\rangle_{t,a} - (\alpha |0\rangle_{b0} + \beta |1\rangle_{b1} )|\psi^-\rangle_{t,a}]
|\psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)
|\phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)
\text{Alice makes a measurement in Bell basis}
Quantum Teleportation Protocol
|\Psi\rangle = |\psi\rangle_t\otimes|\psi^{-}\rangle_{a,b}
=\frac{1}{\sqrt{2}}(\alpha |0\rangle_{t0} + \beta |1\rangle_{t1})\otimes(|01\rangle_{a,b} - |10\rangle_{a,b})
=\frac{1}{2}[(\alpha |1\rangle_{b1} - \beta |0\rangle_{b0} ) |\phi^+\rangle_{t,a} + (\alpha |1\rangle_{b1} + \beta |0\rangle_{b0} )|\phi^-\rangle_{t,a}
+ (\beta |1\rangle_{b1} - \alpha |0\rangle_{b0} )|\psi^+\rangle_{t,a} - (\alpha |0\rangle_{b0} + \beta |1\rangle_{b1} )|\psi^-\rangle_{t,a}]
|\psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)
|\phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)
\text{Bob applies the correct gates to retrieve Alice's qubit}
Deutsch-Josza Algorithm
\text{Oracle } f : \{0,1\}^{n} \rightarrow \{0,1\}
| x | f0 | f1 | f2 | f3 |
|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 |
\text{Is } f \text{ constant or balanced?}
n = 1
Deutsch-Josza Algorithm
Classical: 2 queries
Quantum: 1 query
U_f |x\rangle |y\rangle = |x\rangle |y\oplus f(x)\rangle
Deutsch-Josza Algorithm
U_f |x\rangle |y\rangle = |x\rangle |y\oplus f(x)\rangle
|01\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\text{Apply the Hadamard gates}
Deutsch-Josza Algorithm
U_f |x\rangle |y\rangle = |x\rangle |y\oplus f(x)\rangle
|01\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{2}(|0\rangle |0\oplus f(0)\rangle + |1\rangle |0\oplus f(1)\rangle + |0\rangle |1\oplus f(0)\rangle + |1\rangle |1\oplus f(1)\rangle)
\text{Apply the Oracle gate}
Deutsch-Josza Algorithm
U_f |x\rangle |y\rangle = |x\rangle |y\oplus f(x)\rangle
|01\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{2}(|0\rangle |0\oplus f(0)\rangle + |1\rangle |0\oplus f(1)\rangle + |0\rangle |1\oplus f(0)\rangle + |1\rangle |1\oplus f(1)\rangle)
\text{Reorganize the terms}
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)(-1)^{f(0)}\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
Deutsch-Josza Algorithm
U_f |x\rangle |y\rangle = |x\rangle |y\oplus f(x)\rangle
|01\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{2}(|0\rangle |0\oplus f(0)\rangle + |1\rangle |0\oplus f(1)\rangle + |0\rangle |1\oplus f(0)\rangle + |1\rangle |1\oplus f(1)\rangle)
\text{Discard the second qubit}
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)(-1)^{f(0)}\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)
Deutsch-Josza Algorithm
U_f |x\rangle |y\rangle = |x\rangle |y\oplus f(x)\rangle
|01\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{2}(|0\rangle |0\oplus f(0)\rangle + |1\rangle |0\oplus f(1)\rangle + |0\rangle |1\oplus f(0)\rangle + |1\rangle |1\oplus f(1)\rangle)
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)(-1)^{f(0)}\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)
\rightarrow \frac{1}{2}(|0\rangle + |1\rangle + (-1)^{f(0)\oplus f(1)}|0\rangle - (-1)^{f(0)\oplus f(1)}|1\rangle)
\text{Apply the Hadamard gates}
Deutsch-Josza Algorithm
U_f |x\rangle |y\rangle = |x\rangle |y\oplus f(x)\rangle
|01\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{2}(|0\rangle |0\oplus f(0)\rangle + |1\rangle |0\oplus f(1)\rangle + |0\rangle |1\oplus f(0)\rangle + |1\rangle |1\oplus f(1)\rangle)
\rightarrow \frac{1}{2}(|0\rangle + |1\rangle + (-1)^{f(0)\oplus f(1)}|0\rangle - (-1)^{f(0)\oplus f(1)}|1\rangle)
\rightarrow \frac{1}{2}[(1 + (-1)^{f(0)\oplus f(1)})|0\rangle + (1 - (-1)^{f(0)\oplus f(1)})|1\rangle]
\text{Reorganize the terms}
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)(-1)^{f(0)}\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)
Deutsch-Josza Algorithm
U_f |x\rangle |y\rangle = |x\rangle |y\oplus f(x)\rangle
|01\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{2}(|0\rangle |0\oplus f(0)\rangle + |1\rangle |0\oplus f(1)\rangle + |0\rangle |1\oplus f(0)\rangle + |1\rangle |1\oplus f(1)\rangle)
\rightarrow \frac{1}{2}(|0\rangle + |1\rangle + (-1)^{f(0)\oplus f(1)}|0\rangle - (-1)^{f(0)\oplus f(1)}|1\rangle)
\rightarrow \frac{1}{2}[(1 + (-1)^{f(0)\oplus f(1)})|0\rangle + (1 - (-1)^{f(0)\oplus f(1)})|1\rangle]
\text{if measured } |0\rangle \text{, then } f \text{ is constant, and balanced otherwise}
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)(-1)^{f(0)}\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\rightarrow \frac{1}{\sqrt{2}}(|0\rangle + (-1)^{f(0)\oplus f(1)}|1\rangle)
How do you find a needle
in a haystack?
How do you find a needle
in a haystack?
Unstructured Search
Classical
T(N) = \mathcal{O}(N)
Quantum
T(N) = \mathcal{O}(\sqrt{N})
Unstructured Search
Classical
T(N) = \mathcal{O}(N)
Quantum
T(N) = \mathcal{O}(\sqrt{N})
Our discussion of qubit has been abstract
Our discussion of qubit has been abstract
How can we realize a physical qubit?
p
e-
|0\rangle
Hydrogen
p
e-
|1\rangle
Hydrogen
p
e-
e-
|\psi\rangle
Hydrogen
Particle in a Box
m
0
l
|\psi (t)\rangle = \text{ }?
Schrödinger Equation
i\hbar\frac{d}{dt}|\psi\rangle = \hat{H} |\psi\rangle
Schrödinger Equation
i\hbar\frac{d}{dt}|\psi\rangle = \hat{H} |\psi\rangle
|\psi (t)\rangle = e^{-i\frac{\hat{H}}{\hbar}t} |\psi (0)\rangle
\text{General solution for differential equation}
General Solution
|\psi (t)\rangle = e^{-i\frac{\hat{H}}{\hbar}t} |\psi (0)\rangle
General Solution
|\psi (t)\rangle = e^{-i\frac{\hat{H}}{\hbar}t} |\psi (0)\rangle
|\psi (0)\rangle = \sum_n c_n|\psi_n \rangle
\text{WLOG, initial state is a superposition of arbitrary basis states}
General Solution
|\psi (t)\rangle = e^{-i\frac{\hat{H}}{\hbar}t} |\psi (0)\rangle
|\psi (0)\rangle = \sum_n c_n|\psi_n \rangle
|\psi (t)\rangle = \sum_n c_n e^{-iE_n t/\hbar}|\psi_n \rangle
\text{Plug the second equation into the first equation}
Particle in a Box
|\psi_n\rangle = \sqrt{\frac{2}{l}}\sin{(\frac{n\pi}{l}x)}
E_n = \frac{n^2\pi^2 \hbar^2}{2ml^2}
Particle in a Box
|0\rangle = \sqrt{\frac{2}{l}}\sin{(\frac{\pi}{l}x)}
E_1 = \frac{\pi^2 \hbar^2}{2ml^2}
|1\rangle = \sqrt{\frac{2}{l}}\sin{(\frac{2\pi}{l}x)}
E_2 = \frac{2\pi^2 \hbar^2}{ml^2}
Particle in a Box
|\psi(0)\rangle = \alpha |0\rangle + \beta |1\rangle
|\psi (t)\rangle = \alpha |0\rangle e^{-\frac{iE_1 t}{\hbar}} + \beta |1\rangle e^{-\frac{iE_2 t}{\hbar}}
|\psi (t)\rangle = e^{-\frac{iE_1 t}{\hbar}} (\alpha |0\rangle + \beta |1\rangle e^{-\frac{i(\Delta E)t}{\hbar}})
\Delta E \approx 10 \text{ eV}
\nu = \frac{\Delta E}{h} = 2.5 \times 10^{15} \text{ Hz}
\therefore \text{Atomic qubits are controlled optically via interactions with light pulses}
State-of-the-art
Trapped Atomic Ion
- very long (>>1s) memory
- <20 coherent qubits
- engineering needed
- connection reconfigurable
Superconducting Loop
- short (10^-6s) memory
- <10 coherent qubits
- printable circuits and VLSI
- not reconfigurable
