CSC 591 Quantum Computing Midterm

Pauli X Gate matrix

((0, 1)(1, 0))

Outer product in Braket notation

Given |a〉 and |b〉, inner product = |b〉〈a| = col(α_0, α_1)*row(β_0,β_1)) = matrix((α_0β_0, α_0β_1) (α_1β_0, α_1β_1))

Inner product in Braket notation

Given |a〉 and |b〉, inner product = 〈b|a〉 = row(β_0^,β_1^)⋅col(α_0, α_1) 〈b|a〉 = 〈a|b〉

CNOT matrix

Identity matrix on top left, Pauli X on bottom right

Describe Grover's search method

Involves a diffusion operator that is applied a number of times equal to the square root of the number of qubits to find the target(s); shows quantum advantage in doing an N-element search

Describe D-Wave's factoring method

Involves brute force and therefore does NOT result in any reduction of computational complexity; enumerates all sampled solutions obtained by annealing to provide a pair of factors for a large number

Linearly independent

The vectors are not scalar multiples of each other

How quantum annealing affects a single qubit

The weights (bias) of a qubit changes the state from equal superposition to a ratio a:b of states 0:1

Pauli Z Gate matrix

((1, 0)(0, -1))

Hadamard matrix

((1/sqrt(2), 1/sqrt(2)(1/sqrt(2), -1/sqrt(2))

Penalty function for classical constraint (x + y = 1)

-x - y + 2xy + 1

Penalty function for classical constraint (x + y >= 1)

-x - y + xy + 1

Postulate of Quantum Mechanics

1. Associated to any isolated physical system is a Hilbert space called the state space, which has a unit (state) vector that completely specifies the system 2. Every observable attribute of a physical system is described by an operator that acts on the kets that describe the system 3. When an observable is measured, the only possible outcome will be an eigenvalue of the operator for that observable 4. For a system in state |ψ〉, when an observable is measured, the probability of obtaining eigenvalue λ_j is (〈a_j|ψ〉〗^2, where |a_j〉 is the eigenstate. 5. When an observable is measured, obtaining eigenvalue λ_j, the state of the system is changed to become the corresponding eigenstate |a_j〉. 6. The evolution of a closed quantum system over time is described by a unitary transformation.

ket in Braket notation

A column (vertical) vector with some elements

bra in Braket notation

A row (horizontal) vector, where each element is a complex conjugate of each element in the ket

Basis vectors

A set of vectors that are linearly independent, and that all other vectors in the vector space are linear combinations of vectors in that set

Unitary matrix

A square complex matrix whose adjoint equals its inverse; Given U = matrix((a, b)(c, d)) U-dag = matrix((a*, b*)(c*, d*)), is unitary if U * U-dag = identity matrix

What is the difference between adiabatic quantum computing (AQC) and adiabatic quantum optimization (AQO)?

AQC is a more powerful than AQO; AQO has only one degree of freedom (Z) while AQC has two (Z,X), meaning that ACQ has additional interactions for X

How does the compute capability of adiabatic quantum computing/optimization compare to gate-based quantum computing?

AQC is equivalent to gate-based quantum computing, given universal gates and no noise; AQO is only for optimization problems and does no better than classical in that regard

Tensor product

Also known as Kronecker product in this context; given matrix A with generic elements and matrix B with generic elements, Tensor product is multiplying each element of A with B, in matrix order (i.e. a_00 * B, a_01 * B, a_10 * B, a_11 * B). You will get a higher-order matrix

How quantum annealing creates an initial state

By putting all qubits in superposition

Separable vs. Entangled States

Entangled states cannot be expressed as the tensor product of independent qubit states

True or False: A quantum computing circuit representing an algorithm must be run on a quantum computer many times so that the resulting output will only display one unique result from among all of the sampled outcomes.

False; that would be a for a quantum annealing machine

True or False: If a quantum computing circuit is properly initialized so that it is unitary at the outset, the axioms of quantum mechanics guarantee that the dynamical evolution of this system from the initialization to final readout will result in a unitary output when run on today's quantum computer hardware platforms.

False; the system can experience a number of gates so that the output isn't unitary. This happens a lot of the time

True or False: To obtain an expectation value of an observable, one must add all of the probability amplitudes to get the final measured result.

False; this is because if one probability amplitude was 1/2 and the other was -1/2, then you would end up with probability amplitude of 0, which doesn't make sense

True or False: In order to view the intermediate steps in a quantum computing calculation, the calculation must be run many times. For each time that the calculation is run it must be evaluated at a different intermediate stopping point so that all of these different results can be pieced together to show the unique explicit full evolution of the quantum computing calculation from initialization to final readout.

False; with quantum computing, one calculation would be irreversibly affected if a previous calculation once was measured

True or False: Step by step evolution from an initial state on a quantum computer to some final state must maintain the initial probability amplitude through to the final measurement.

False; you want to change the initial probability through step by step evolution so that you can measure something meaningful from that process

Hermitian matrix

Given U = matrix((a, b)(c, d)) U-dag = matrix((a*, b*)(c*, d*)), is hermitian if U = U-dag

Describe Shor's factoring method

Involves quantum Fourier transform to find the period of a sequence of some (systematically tried) number that does not divide the two prime factors in question; goes from exponential to logarithmic

SWAP matrix

Like a unitary, but the middle is CNOT

How quantum annealing affects a pairs of qubits on the D-Wave

Strength/coupling between qubits indicates the factional likelihood to align states with one another (positive means more aligned, negative means less aligned)

What remains the same in unitary transformation?

The norm of the target for the transformation

True or False: For a quantum mechanical system, once a measurement is done on the system, all information prior to that measurement is permanently lost.

True

True or False: Hermitian operators generate eigenstates that are always mutually orthogonal.

True

True or False: The probability amplitude has an indeterminate specific value until a measurement is performed.

True

Hilbert space

vector space over complex numbers with an inner product ⟨b|a⟩

Penalty function for classical constraint (x = y)

x + y - 2xy

Penalty function for classical constraint (x <= y)

x - xy

Penalty function for classical constraint (x + y <= 1)

xy

Penalty function for classical constraint (x + y + z <= 1)

xy + xz + yz

Other important quantum states as vectors

|+〉 = (1/sqrt(2), 1/sqrt(2)) |-〉 = (1/sqrt(2), -1/sqrt(2))

Standard basis

|0〉 = (1, 0) |1〉 = (0, 1) Note: ket denotes column vectors