Quantum Circuits

• Quantum circuits are models of quantum computations
  • Wires represent information transfer (qubits)
    • Double lines represent classical information
  • Gates represent operations (unitary operations and measurements)
• In this context, circuits are depicted acyclically (left-to-right information flow)
• Quantum operations are denoted in reverse order
• For convention, ordering qubits from bottom-to-top is equivalent to ordering qubits from left-to-right
• When we have standard basis inputs, each column of the matrix (the product of all unitary operations in a circuit) represents an output given a standard basis combination
• Arbitrary unitary operations can be viewed as gates
  • A single operation on multiple qubits can be represented as a large rectangle

Inner Products

• A ket is a columnvector and a bra is a row vector
  • A bra associated with a ket is the ket’s conjugate transpose
• The inner product $⟨\psi ||\varphi ⟩$ is a complex scalar
• Geometrically, the inner product of two unit vectors with real entries is the cosine of the angle between them ← if inner product is zero, the vectors are orthogonal
• Properties of inner products
  • The square root of the inner product of a vector with itself is the Euclidean norm
  • Inner products exhibit conjugate symmetry (swapping the vectors conjugates the inner product)
  • Linearity in the second argument → $⟨\psi ||\varphi ⟩={\alpha }_1⟨\psi ||{\varphi }_1⟩+{\alpha }_2⟨\psi ||{\varphi }_2⟩$
  • Conjugate linearity in the first argument
  • Inner products exhibit the Cauchy-Schwarz inequality → $|⟨\psi ||\varphi ⟩|\le |||\psi ⟩|||||\varphi ⟩||$

Orthogonality and Orthonormality

• Two vectors are orthogonal if their inner product is zero
• An orthogonal set is the set of all vectors where any pair of them is orthogonal
• An orthonormal set is an orthogonal set whose members are all unit vectors
  • Orthonormal sets are always linearly independent
  • If m < n (m is the number of vectors in the set, and n ← n-dimensional space from which the vectors are drawn), we can add more vectors to the set to make the entire set an orthonormal basis from m+1 to n
  • The Gram-Schmidt orthogonalization process can be used for this
• An orthonormal basis is an orthonormal set that forms the basis of a space
  • $\{|+⟩,|-⟩\}$ is the orthonormal basis for the 2D space corresponding to a single qubit
  • The Bell basis is the orthonormal basis for the 4D space corresponding to two qubits
• The following three conditions are equivalent w.r.t unitary matrices
  • A unitary matrix U is when the product of itself and its conjugate transpose results in the identity matrix
  • The rows of U form an orthonormal basis
  • The columns of U form an orthonormal basis

Equivalence

A set of statements are equivalent if all of them are either true or false together

Projections

• A square matrix $\Pi$ is a projection if
  • $\Pi$ is hermitian ($\Pi ={\Pi }^{†}$)
  • ${\Pi }^2$ = $\Pi$
• Intuitively, a projection operation on a vector leaves behind one component of that vector while zeroing out the rest
  • Applying the projection on a vector on which a projection was already applied doesn’t do anything because the other components are already removed. This is mathematically represented as ${\Pi }^2=\Pi$
• A projective measurement is a collection of projections whose net sum is the identity matrix
• When a measurement is performed
  • Pr(outcome is k) = ||${\Pi }_k|\psi ⟩$||${}^2$ → The outcome k is chosen randomly
  • The state of the system being measured is $\frac{{\Pi }_k|\psi ⟩}{||{\Pi }_k|\psi ⟩||}$
• Standard basis measurements are projective measurements
• Projective measurements can be implemented in quantum circuits using unitary matrices and standard basis measurements

Limitations on Quantum Information

• Irrelevance of global phases
  • Two quantum states vectors differ by a global phase when one of them = the other multiplied by a complex number
    • The absolute value of the complex number is 1
    • The probabilites of measuring an outcome a will be the same with both vectors. Same for projective measurements
  • The two vectors that differ by a global phase are completely indistinguishable and are equivalent
  • Vectors that do not differ by a global phase are
    • Linearly dependent
    • Have a relative phase difference
    • Can be perfectly distinguishable
• No-cloning theorem
  • It is impossible to create a perfect copy of an unknown quantum state
  • For any quantum state $|\psi ⟩$, there is no unitary operation that can clone a quantum state. U on the state tensored with zero will not give psi tensored with psi
  • The transformation is non-linear but unitary operations are linear. This proves the no-cloning theorem
  • Approximate copies can be done
  • Copying a standard basis state is possible and the no-cloning theorem doesn’t contradict it
  • Cloning a probabilistic state (classically) is also impossible
• Discriminating non-orthogonal states
  • It isn’t possible to perfectly discriminate two non-orthogonal quantum states
    • If they can be perfectly distinguishable, they have to be orthogonal