2. What is a quantum computer? A quantum computer is any device for computation that makes use of quantum mechanical phenomena to perform operations on data, implemented with quantum binary digits (qubits) in particle spin states. What are the advantages? Search of unstructured data base with N items would take N steps on Classical Computer and only √ n steps on a quantum computer. (Grover algorithm) Quantum computers can perform parallel computations on superposed data. - Shor’s algorithm (Factorization with Fourier transform) Q uantum C omputer O verview Quantum computers are highly efficient for:  Factoring large numbers!  Simulation of quantum systems!  Solving discrete logarithms!

3. Quantum Registers, Ket Notation StateAmplitudeProbability - a+bia 2 +b 2 000.37+.04i.14 001.11+.18i.04 010.09+.31i.10 011.30+.30i.18 100.35+.43i.31 101.40+.01i.16 110.09+.12i.02 111.15+.16i.05 Adds up to 1! Example: (With arbitrary amplitudes – |a+bi| is √(a 2 +b 2 )) Sign affects relative phase factor Where |α| 2 + |β| 2 = 1 and |α| 2 or |β| 2 give the probability of observing each state. Since all possible 2 n binary states of an n-qubit quantum register can be superposed, the register can be described by a state vector: |Ψ> = α 0 |…000> + α 1 |…001> + α 2 |…010> + α 3 |…011> + … + α n |?> …or in a column with 2n entries correlating binary states to amplitudes. Ψ 1 = α 1 |0> ± β 1 |1> Typical state vector of single-particle system: Complex Amplitudes Eigenstates

4. Quantum Array Notation vs. Classical All quantum circuits are reversible (have a one-to-one correspondence between input and output vectors), so quantum array gates do not have an implicate direction like standard Boolean gates. = Product operation = Wire Example (Toffoli EXOR-middle): = EXOR operation a c b P=a R=c Q=acb

5. Quantum Permutative Gates Overview All quantum gates are represented by unitary matrices that define transformations on qubits. Permutative gates are gates whose unitary matrices are simply permutative matrices. Five major types and their tautological equivalents: -Inverter (NOT – also a quantum primitive) -Feynman (Controlled-NOT) -Toffoli (Controlled-Controlled-NOT) -Fredkin (Controlled-Swap) -Swap These gate-matrices can also be described by how many inputs are fed to the output with no transformations, in which case they are said to be “k-through.” A reversible gate with n inputs/outputs is described by a 2 n * 2 n matrix.

6. Inverter (NOT) Logic: Schematic: (zero-through) Unitary Matrix: P=a 0 1 1 0 0 1 0 1 X aP

7. Feynmann (CNOT) Unitary Matrices:Logic:Schematic: (one-through) (P, Q)=(a, a b) (P, Q)=(a b, b) a bQ P a bQ P 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 00 01 10 11 00 01 10 11 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 00 01 10 11 00 01 10 11 EXOR-Down EXOR-Up

8. Toffoli (CCNOT) Unitary Matrix: Logic: Schematic of EXOR-down: (two-through) a bQ P cR (P, Q)=(a, b), R=ab c 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 0 000 001 010 011 100 101 110 111 000 001 010 011 100 101 110 111

9. Fredkin (CSWAP) Unitary Matrix: Logic: Schematic of Fredkin-Up: (one-through) a bQ P cR P=a, Q=if (a=1) then c else b, R= if (a=1) then b else c P=a, Q=ab ac, R=ac ab or 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 0 0 0 0 0 0 0 0 1 000 001 010 011 100 101 110 111 000 001 010 011 100 101 110 111

10. Swap Gate Unitary Matrix: Logic: Schematic: (zero-through) a bQ P 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 00 01 10 11 00 01 10 11 P=b, Q=a Feynmann Gate Realizations: a bQ P a bQ P

11. Universal Controlled Gate P=a; if a=0 then Q=b; if a=1 then Q=U(b) U a bQ P ?

12. Matrices Overview -Identity Matrix (Square matrix with “1” in entries along main diagonal, “0s” elsewhere): Testing for Unitary Matrix U: (Example: Phase gate matrix) 1) Take the hermitian transpose U + of U: 2) Verify that U * U + = I (the corresponding n * n identity matrix): -Permutation Matrix: Square matrix with entry of “1” in each row/column I n = 1 0 0 0 1 0 0 0 1 … … … ……… … -Inverse Matrix: Square matrix A -1 of A such that AA -1 =A -1 A=I n -Orthogonal Matrix: Square matrix Q whose transpose is its inverse: QQ T =Q T Q=I n a b c d Where: 1) a 2 +c 2 = 1 2) b 2 +d 2 = 1 3) ab+cd = 0 1 0 0 i + = 1 0 0 -i 1 0 0 1 1 0 0 -i 1 0 0 i * (1+0) (0+0) (0+0) (0+1) = ==I2I2

13. Standard Product The product A * B of two quantum gate-matrices A and B in a serial connection is used to derive their collective unitary matrix. Example: a bQ P Feynmann-DownSwap = = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 * 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 00 01 10 11 00 01 10 11 Unitary Matrix of Circuit (Permutative)

14. Kronecker Product The Kronecker product A B of two quantum gate-matrices A and B in a parallel connection is used to derive their collective unitary matrix. Example: == 0 -i i 0 0 1 1 0 YX (NOT) 0 0 0 -i 0 0 i 0 0 -i 0 0 i 0 0 0 00 01 10 11 00 01 10 11 Unitary Matrix of Circuit Pauli-X gate Pauli-Y gate a P X Y bQ

15. Matrix Vector A vector is a line segment with a direction and magnitude spanning some set of axes. In quantum mechanics, a complex vector space of n dimensions (axes) is denoted C n (Hilbert space). Column Notation for Vectors: (With entries representing coordinates of the terminal point in complex vector space) The ket represents a vector |u> = u1unu1un … (Corresponds to C n where u n = a n +b n i) Vector Addition: |u> + |v> = u 1 + v 1 u n + v n … Dual/Adjoint Vector : (Transpose column and conjugate entries) † = [u 1 *, …, u n *] y x z

16. Truly Quantum Primitives Most truly quantum primitive gates cannot be decomposed into simpler matrix operators. They include complex entries to reflect state vector amplitudes rather than only standard bits, thus introducing quantum phenomena into reversible circuits (and can also transform qubit inputs into superposed states). HXYZ S V HadamardPauli-X (NOT)Pauli-Y Pauli-ZPhaseV ( √ NOT)

17. Hadamard, Phase, V (√NOT) H 1 1 -1 1 √2 Unitary Matrices: S 1 0 0 i V 1+i 1-i 1-i 1+i 1 2

18. Pauli- X (NOT), Y, Z Unitary Matrices: Z Y X 0 1 1 0 0 -i i 0 1 0 0 -1

19. Simple Circuit Equivalences = HH = UU+U+ = VVX 1 2 1+i 1-i 1-i 1+i 2 = 0 1 1 0 1 √2 1 1 -1 2 = 1 0 0 1 1 2 2 0 0 2 = U * U + = I

20. Karnaugh Maps and Testing In testing for constant or balanced Boolean functions, 2 tries is the best-case scenario for a regular computer while n/2+1 tries is the worst (for a function with n possible output bits). A quantum computer requires only one step using the Deutsch-Jozsa algorithm. Truth Table (Displays input/output function) Karnaugh Map (Displays input/output “coordinates”) EXOR Operation: A B P 0 0 0 0 1 1 1 0 1 1 1 0 A 01 01 0 1 0 1 B Balanced Function f(n): A 00 01 10 11 B 0 1111 1111 000 0000 00 01 10 11 A 00 01 10 11 B 0 1111 1111 000 0000 00 01 10 11

21. Bloch Sphere!! *On the Bloch sphere, a qubit’s state vector is described as |Ψ> = cosθ |0> + e iφ sinθ |1> for –π/2 ≤ θ < π/2 and 0 ≤ φ < 2π *The corresponding Bloch vector’s terminal point on the sphere’s surface is given by the coordinates: xyzxyz = sin 2θ × cos φ sin 2θ × sin φ cos 2θ Z YX 0 1 1 1 |0 › NOT √ NOT Ψ |1 › 2θ2θ φ

22. Ternary Logic Ternary logic uses three basic bits: 0, 1, and 2. Ternary quantum registers can have 3 n superposed states for n-qubit inputs. -Ternary NOT gates include +1, +2, (01), (02), and (12) with parentheses denoting permutations on single-bit inputs (in addition to equivalent √NOT gates) Thus, the state vector would be: Ψ = α|0> + β|1> + γ|2> where |0> = 100100 (0° rotation of axis on qubit) |1> = 010010 (120° rotation of axis on qubit) |2> = 001001 (240° rotation of axis on qubit) Chrestenson Gate = basic ternary NOT (where a=e i(2π/3) ): 1 1 1 1 a a 2 1 a 2 a CH =

23. HΨ(x,t) = iħ[dΨ(x,t)/dt] Design Realizations Criteria: (DiVincenzo checklist) Proposed Quantum Computer Architectures:  Superconductor arrays  Ion traps  Nuclear magnetic resonance (NMR) on solutions of molecules  Solid-state NMR  Quantum dot surfaces  Cavity quantum electrodynamics (CQED) structures  Molecular magnets Phenomena: -Superposition: “Blend” of eigenstates in QM system (QM database searching) -Entanglement: Instantaneous particle correlation (QM teleportation, communication) -Interference: Disruption of QM systems (QM parallelism, joint computations) -Non-clonability: Impossible to copy unknown QM state (QM cryptography) Physically scalable (qubits can be increased) Qubits can be initialized to some values Operates much faster than decoherence time Computer uses quantum gates and logic Has a means of reading qubits

24. Conclusion “If you put crap into a computer, nothing comes out of it but crap. But this crap, having passed through a very expensive machine, is somehow enobled and no-one dares criticize it.” -Pierre Gallois