Introduction to Quantum Computing Week 1

Scope of mechanics

Classical mechanics (Newton’s laws, electronic current => classical computing) rules medium scale world, while Quantum mechanics (Entanglement, tunnelling, quantum superposition…) rules the world of atoms and molecules.

Quantum computing can solve some some hard problems more efficiently compared with classic mechanics computing.

Classical computing and Quantum computing

A classical bit, a semiconductor in chip, is like a switch that could be turned on (represented as 1) or off (represented as 0). Here in classical mechanics, a bit can only be in one status, that is either 1 or 0. We cannot accept a ‘super status’ of a light switch that is both on and off at the same time. With this format, we can represent data in a way of binary format, e.g. (10 represents 2, 11 represent 3 and so on) formally $X = \Sigma_{i}^{n}d_i2^i$

But in quantum context, we can have different status in a quantum bit, or ‘qubit’ called ground state (n=1) and excited state (n=2). What different is that a qubit can be both 0 and 1 (superpositions) before a measurement or observation, represented as $\alpha \rvert 0 \rangle + \beta \rvert 1\rangle$, where $\alpha$ and $\beta$ are called amplitudes and are complex numbers.

If we were to measure or observe, the probability of zero is $\lvert \alpha \rvert^2$, and these probabilities must sum to 1

\[\lvert \alpha \rvert^2 + \lvert \beta\rvert^2 = 1\]

By combining multiple qubits, we can have binary representation like $\rvert 00\rangle$, $\rvert 01\rangle$, $\rvert 10\rangle$ etc

The state of a quantum computer is a super position over all binary strings, k,

\[\rvert \psi\rangle = \sum_k \alpha_k\rvert k \rangle\]

where $\alpha_k$ are amplitudes and $k$ are states.

Although they have superpositions, it can only collapses to one random N-bit string when a measurement is taken. Before measurement, we can get the probability of different state in superpositions. In other words, the amplitudes tell us how much of $\rvert 0 \rangle$ and $\rvert 1 \rangle$ are in the superposition $\rvert\psi\rangle$, and the modulus squared of the amplitudes tell us the probability. e.g. $\rvert\psi\rangle = 0.775\rvert 0 \rangle + 0.633 \rvert 1 \rangle$ tells us that the probability the outcome is 1 is $0.633^2=0.4$ while the probability the outcome is 0 is $0.775^2=0.6$.

How do we handle the probability and the ‘noise’?

• Noisy Intermediate Scale Quantum, NISQ
  is a medium scale system with 10~1000 qubits that have simplified control and error correction. Potentially polynomial speed-up.
• Fault tolerant quantum computing is a large-scale system with far more than 1000 qubits, which is high redundancy for error correction and have potentials to exponentially speed-up for some problems.

Math prep

1. Complex number $z = x+iy$ where Real part is denoted by $\text{Re}[z]=x$ and the Imaginary part is denoted by $\text{Im}[z]=y$.
   • Conjugate: $z = x+iy \rightarrow z^*=x-iy$
   • $zz^*=x^2+y^2 = z^2$, hence $\lvert z \rvert = \sqrt{x^2+y^2}$
   • $\lvert z \rvert$ is referred as the modulus or magnitude.

   

2. Different notation
   • Cartesian: $z = x+iy$
   • Polar: $z = r(\cos\theta + i\sin\theta)=re^{i\theta}$
   • NB identity: $e^{i\theta}=\cos\theta + i\sin\theta$, $r=\lvert z\rvert$

   For Polar notation, if $a=\text{Re}[a]+i\text{Im}[a]=\lvert a\rvert e^{i\theta}$, we have $\lvert a\rvert = \sqrt{\text{Re}[a]^2+\text{Im}[a]^2}$ and $\theta = \tan^{-1}(\text{Im}[a]/\text{Re}[a])$

   For each complex amplitude, we have a magnitude and phase angle: $\rvert\psi\rangle = \alpha_0\rvert 0 \rangle + \alpha_1\rvert 1\rangle$ $\rightarrow$

   \[\rvert\psi\rangle = \rvert a_0\rvert e^{i\theta_0}\rvert 0\rangle + \rvert a_1\rvert e^{i\theta_1}\rvert 1\rangle\]

3. $\rvert\psi\rangle$ is a ‘ket’, an element of a linear vector space over $C$ which represents the state of a qubit. We write the general state of a qubit in “ket” notation as:

   \[\rvert\psi\rangle = a_0\rvert 0 \rangle + a_1\rvert 1 \rangle\]

   A pure state, could be represented as

   \[\rvert 0\rangle =\bigl(\begin{smallmatrix} 1\\ 0 \end{smallmatrix}\bigr)\] \[\rvert 1\rangle =\bigl(\begin{smallmatrix} 0\\ 1 \end{smallmatrix}\bigr)\]

   and $a_0\rvert 0 \rangle + a_1\rvert 1 \rangle$ could be represented as:

   \[\rvert\psi\rangle =\bigl(\begin{smallmatrix} a_0\\ a_1 \end{smallmatrix}\bigr)\]

   We define a ‘bra’ $\langle\psi\rvert$ as a dual vector to be:

   \[\langle\psi\rvert = [a_0^* \; a_1^*]\]

4. basic operation:

   • inner product of $\vert\psi\rangle=[a\;b]^{-1}$ and $\rvert\phi\rangle = [c\;d]^{-1}$

     \[\langle\psi\vert\phi\rangle = a^*c+b^*d\]

   • Orthogonality:

     \[\langle\psi\vert \vert\phi\rangle = 0\]

     Z-basis orthogonal: $[1\;0][0\;1]^{-1} = 0$

     X-basis orthogonal: $[1\;1][1\;-1]^{-1} = 0$

     

   • Outer Product of $\vert\psi\rangle\,\langle\phi\vert$ For two quantum state $\vert\psi\rangle = [a\;b]^{-1}$ and $\vert\phi\rangle = [c\;d]^{-1}$, We can define outer (tensor) product:

     

   • Matrix Transpose
   • Matrix Adjoint To find the “Hermitian adjoint”(or “conjugate transpose”) of a matrix, exchange rows and columns and also take the complex conjugate.
   • Unitary Matrices: All the operations on a quantum computer will be unitary.
   • Hermitian Matrices: $A^\dagger=A$. All their eigenvalues are real.
   • Bra-ket Notations: