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<h1>Quantum computing 1</h1>
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<p><i>November 11, 2020 — Support my next blog post, <a href="https://www.paypal.com/donate/?hosted_button_id=UCLCSJLFL433E">buy me a coffee</a> ☕.</i></p>
<p>In my first research project as an undergraduate student in the summer of 2010, I worked on groups and symmetries in classical and quantum physics. One day in the library I came across <a href='https://en.wikipedia.org/wiki/Noether%27s_theorem'>Noether's theorem</a>, which links symmetries (e.g., time invariance) to conserved quantities (e.g., energy). Noether was one of the leading mathematicians of her time and made major contributions to both <a href='https://en.wikipedia.org/wiki/Algebra'>algebra</a> and <a href='https://en.wikipedia.org/wiki/Mathematical_physics'>mathematical physics</a>. Learning about her work was a decisive moment in my life because I realized for the first time how deeply connected mathematics and physics were.</p>
<p>Thanks to Noether, I got really interested in mathematical physics and took classes in <a href='https://en.wikipedia.org/wiki/Statistical_mechanics'>statistical mechanics</a> and <a href='https://en.wikipedia.org/wiki/Special_relativity'>special relativity</a>. My second research project as an undergraduate student was on <a href='https://en.wikipedia.org/wiki/Riemannian_geometry'>Riemannian geometry</a> and <a href='https://en.wikipedia.org/wiki/General_relativity'>general relativity</a>, with my friend and former classmate <a href='https://www.linkedin.com/in/olivier-martre-7a7a9a95/'>Olivier Martre</a>—oh dear, how hard it was!</p>
<p>A couple of years down the road, I had to choose between an M.Sc. in applied mathematics and an MS.c. in mathematical physics. I ended pursuing a career in applied mathematics, but physics never really left my life—most of my work in applied mathematics has been on the numerical solution of the <a href='https://en.wikipedia.org/wiki/Differential_equation'>differential equations</a> of physics. In particular, during my Ph.D. at Oxford, I built a partial differential equation solver called <code>spin</code>; the guide can be found <a href='https://www.chebfun.org/docs/guide/guide19.html'>here</a>. While it stands for "stiff PDEs integrator", I named it after the <a href='https://en.wikipedia.org/wiki/Spin_(physics)'>quantum mechanical spin</a>. The <code>spin</code> code is part of the MATLAB-based package <a href='https://www.chebfun.org/'>Chebfun</a>, a project led by my former Ph.D. advisor <a href='https://en.wikipedia.org/wiki/Nick_Trefethen'>Nick Trefethen</a>.</p>
<h2>Quantum states and observables</h2>
<p>In <a href='https://en.wikipedia.org/wiki/Quantum_mechanics'>quantum physics</a>, the <a href='https://en.wikipedia.org/wiki/Quantum_state'>state</a> of a physical system is described by its <a href='https://en.wikipedia.org/wiki/Wave_function'>wave function</a> \(\psi\), an element of a <a href='https://en.wikipedia.org/wiki/Separable_space'>separable</a> <a href='https://en.wikipedia.org/wiki/Hilbert_space'>complex Hilbert space</a> \(X\) <a href="#section1">[1]</a>. Physical quantities are represented by <a href='https://en.wikipedia.org/wiki/Self-adjoint_operator'>self-adjoint operators</a> \(L:X\rightarrow X\) acting on \(\psi\), called <a href='https://en.wikipedia.org/wiki/Observable#Quantum_mechanics'>observables</a>. Note that \(X\) may or may not be finite-dimensional; in finite dimension, \(L\) is a <a href='https://en.wikipedia.org/wiki/Hermitian_matrix'>Hermitian matrix</a>. Examples of operators include the <a href='https://en.wikipedia.org/wiki/Position_operator'>position operator</a> (position), the <a href='https://en.wikipedia.org/wiki/Momentum_operator'>momentum operator</a> (momentum), the <a href='https://en.wikipedia.org/wiki/Spin_(physics)'>spin angular momentum operator</a> (spin) and the <a href='https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)'>Hamiltonian operator</a> (total energy).</p>
<p>Let us first look at a finite-dimensional example by considering the spin operator along the \(y\)-axis for <a href='https://en.wikipedia.org/wiki/Spin-%C2%BD'>spin-\(\frac{1}{2}\)</a> particles, such as the <a href='https://en.wikipedia.org/wiki/Electron'>electron</a>. In this case, the quantum state \(\psi=(\psi_1,\psi_2)^T\), with respect to \(y\)-spin, lives in is the Hilbert space \(X=\mathbb{C}^2\) of pairs of complex numbers, with inner product
$$
\langle\psi,\phi\rangle = \overline{\psi}_1\phi_1 + \overline{\psi}_2\phi_2,
$$
and the spin operator along the \(y\)-axis is the matrix \(L=S_y\) given by
$$
S_y = \frac{\hbar}{2}\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}.
$$
Since it is a Hermitian matrix, the <a href='https://en.wikipedia.org/wiki/Spectral_theorem'>spectral theorem</a> guarantees that it is diagonalizable with <a href='https://en.wikipedia.org/wiki/Orthonormality'>orthonormal</a> eigenvectors and real eigenvalues. The eigenvalues are \(\pm\frac{\hbar}{2}\), while the eigenvectors are given by
$$
\psi_{y+}=\frac{1}{\sqrt{2}}\begin{pmatrix}
1 \\
i
\end{pmatrix}, \quad
\psi_{y-}=\frac{1}{\sqrt{2}}\begin{pmatrix}
1 \\
-i
\end{pmatrix}.
$$
The quantum state of a spin-\(\frac{1}{2}\) particle, with respect to \(y\)-spin, may then be written as
$$
\psi = \langle\psi_{y+},\psi\rangle\psi_{y+} + \langle\psi_{y-},\psi\rangle\psi_{y-},
$$
with the normalization condition
$$
\langle\psi,\psi\rangle = \vert\langle\psi_{y+},\psi\rangle\vert^2 + \vert\langle\psi_{y-},\psi\rangle\vert^2 = 1.
$$
When the \(y\)-spin is measured, the probability that it will be \(\pm\frac{\hbar}{2}\) is \(\vert\langle\psi_{y\pm},\psi\rangle\vert^2\). Following the measurement, \(\psi\) will collapse into the corresponding eigenstate. Note that the <a href='https://en.wikipedia.org/wiki/Expected_value'>expected value</a> of \(\psi\) is given by
$$
\langle\psi\rangle = \frac{\hbar}{2}\vert\langle\psi_{y+},\psi\rangle\vert^2 - \frac{\hbar}{2}\vert\langle\psi_{y-},\psi\rangle\vert^2.
$$
</p>
<p>Let us now look at an example in infinite dimension by considering the Hamiltonian operator of a particle of mass \(m\) in a quadratic potential \(V(x)=\frac{1}{2}m\omega^2x^2\), the so-called <a href='https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator'>quantum harmonic oscillator</a> with frequency \(\omega\). In this case, the quantum state \(\psi\), with respect to total energy, belongs to the Hilbert space \(X=L^2(\mathbb{R})\) of <a href='https://en.wikipedia.org/wiki/Square-integrable_function#:~:text=A%20space%20which%20is%20complete,induced%20by%20the%20inner%20product.'>square-integrable functions</a> \(\psi\), with inner product
$$
\langle\psi,\phi\rangle = \int_{\mathbb{R}}\overline{\psi}(x)\phi(x)dx,
$$
and the Hamiltonian operator is the operator \(L=H\) defined by
$$
H\psi(x) = -\frac{\hbar}{2m}\psi''(x) + \frac{1}{2}m\omega^2x^2\psi(x), \;\;x\in\mathbb{R},
$$
with domain <a href="#section1">[2]</a>
$$
D(H) = \big\{\psi\in H^2(\mathbb{R})\,|\,x^2\psi\in L^2(\mathbb{R})\big\}\subset L^2(\mathbb{R}).
$$
It is an example of an <a href='https://en.wikipedia.org/wiki/Unbounded_operator'>unbounded operator</a>, since the elements of its spectrum have unbounded amplitude. It is diagonalizable with orthonormal eigenfunctions and a discrete real spectrum <a href="#section1">[3]</a>. The countably many eigenvalues are given by
$$
\lambda_n = (2n+1)\frac{\hbar}{2}\omega, \quad n=0,1,2,\ldots,
$$
while the eigenfunctions are scaled <a href='https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_functions'>Hermite functions</a> \(\psi_n\).
Therefore, one may write the quantum state of the particle, with respect to total energy, as
$$
\psi(x) = \sum_{n=0}^{\infty}\langle\psi_n,\psi\rangle\psi_n(x),
$$
with the normalizaton condition
$$
\langle\psi,\psi\rangle = \sum_{n=0}^{\infty}\vert\langle\psi_n,\psi\rangle\vert^2 = 1.
$$
As in the finite-dimensional case, the meaning of the decomposition in the basis of eigenfunctions is that if we perform a total energy measurement, the state will have total energy \(\lambda_n\) with probability \(\vert\langle\psi_n,\psi\rangle\vert^2\), and following the measurement, the state will collapse into the corresponding eigenstate. Here, the expected value of \(\psi\) is given by
$$
\langle\psi\rangle = \sum_{n=0}^{\infty}\lambda_n\vert\langle\psi_n,\psi\rangle\vert^2.
$$
In Chebfun, quantum states can be computed with the <code>quantumstates</code> command; see, e.g., <a href='https://www.chebfun.org/examples/ode-eig/Eigenstates.html'>this example</a>.
</p>
<h2>Bit versus qubit</h2>
<p>Classical computers use <a href='https://en.wikipedia.org/wiki/Bit'>bits</a> to store information, which are commonly represented as "0" and "1". The way I like to think about bits is that each bit is like a small magnet that can either point North ("0") or South ("1"). By applying an appropriate magnetic field, one can change the magnetization of a bit, and change "0" to "1", or vice versa. Bits are then combined together into arrays of bits, called words. Numbers, for example, are stored as 32- or 64-bit words, following the <a href='https://en.wikipedia.org/wiki/IEEE_754'>IEEE 754</a> technical standard.</p>
<p>Quantum computers use <a href='https://en.wikipedia.org/wiki/Qubit'>qubits</a>. A qubit is a two-state quantum-mechanical system, for example, the \(y\)-spin of the electron described in the previous section. In that scenario, a qubit would be
$$
\psi = \alpha\psi_{y+} + \beta\psi_{y-},
$$
where \(\psi_{y+}\) corresponds to "0" and \(\psi_{y-}\) to "1". As in the classical case, one can change the value of a qubit by applying an appropriate magnetic field. Note that a qubit does not have a value in-between "0" and "1"; rather, as explained previously, when measured, it has a probability \(\vert\alpha\vert^2\) of the value "0" and a probability \(\vert\beta\vert^2\) of the value "1".
</p>
<p>In the next post, I will talk about the <a href='https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation'>Schrödinger equation</a> and <a href='https://en.wikipedia.org/wiki/Quantum_logic_gate'>quantum logic gates</a>.</p>
<p id="section1">[1] To be more precise, one has to consider equivalent classes of elements in \(H\) called <a href='https://en.wikipedia.org/wiki/Projective_Hilbert_space'>rays</a>, with \(\psi\sim\phi\) when \(\phi=\lambda\psi\) for some non-zero \(\lambda\in\mathbb{C}\).
<p id="section1">[2] \(H^2(\mathbb{R})\) denotes the usual <a href='https://en.wikipedia.org/wiki/Sobolev_space#The_case_p_=_2'>Sobolev space</a> of functions in \(L^2(\mathbb{R})\) whose weak derivatives up to order two have a finite \(L^2(\mathbb{R})\)-norm.
<p id="section1">[3] Self-adjoint unbounded operators can always be diagonalized in some sense but, in general, the spectrum has both a discrete and a continuous part; see, e.g., <a href='https://en.wikipedia.org/wiki/Self-adjoint_operator#Spectral_theorem'> this.</a>
<hr>
<h4>Blog posts about quantum computing</h4>
<p>2020 <a href="2020-12-06.html">Quantum computing 4</a></p>
<p>2020 <a href="2020-11-28.html">Quantum computing 3</a></p>
<p>2020 <a href="2020-11-17.html">Quantum computing 2</a></p>
<p>2020 <a href="2020-11-11.html">Quantum computing 1</a></p>
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