update first week

Author: mhjensen

doc/pub/week1/html/week1-bs.html

@@ -551,6 +551,34 @@ <h2 id="the-options">The options </h2>
 <p>At least one of these must be true!</p>
 </section>
 
+<section>
+<h2 id="naive-yet-popular-statement">Naive yet popular statement  </h2>
+
+<p>A naive, yet popular, statement says that quantum computers can
+provide an exponential speedup over classical computers because their
+internal states live in an exponentially large space of dimension
+\( 2^N \) for an \( N \)-qubit quantum computer. The number of dimensions
+grows so fast with \( N \) that classical supercomputers cannot even hold
+this state in memory as soon as \( N>50 \); hence, classical computing is
+supposedly doomed to address these states.
+</p>
+</section>
+
+<section>
+<h2 id="more-to-the-picture-than-meets-the-eye">More to the picture than meets the eye </h2>
+
+<p>Yet, despite this supposed impossibility, a rather large number of
+such exponentially large states have been calculated, sometimes with
+machine precision, using classical algorithms.
+</p>
+
+<p><b>The solution of this small paradox is the
+same as in other successes of physics: apparently very complex
+phenomena have internal mathematical structures that, when revealed,
+allow one to make precise predictions.</b>
+</p>
+</section>
+
 <section>
 <h2 id="present-day">Present day </h2>
 <p>Quantum computing is an extremely active and exciting field:</p>
@@ -891,6 +919,258 @@ <h2 id="quantum-platforms">Quantum platforms </h2>
 </div>
 </section>
 
+<section>
+<h2 id="quantum-computing-in-a-nutshell">Quantum computing in a nutshell </h2>
+
+<p>A quantum computer is a well-controlled out-of-equilibrium quantum
+many-body system that one intends to use to perform a
+calculation. Such a system can be described at several levels: from
+the actual underlying physics (usually described in terms of its
+Hamiltonian, that is with time and energies) up to an abstract
+representation used to describe quantum algorithms, the so-called <b>gate-based
+quantum computer</b>.
+</p>
+</section>
+
+<section>
+<h2 id="an-exponentially-large-internal-state">An exponentially large internal state </h2>
+
+<p>The abstract <b>gate-based</b> quantum computer is defined as follows. We
+have a set of \( N \) two-level systems, called quantum bits or qubits
+(for instance, the spin of an electron), that can be in the states
+\( \vert 0\rangle \) and \( \vert 1 \rangle \). The most general state of the quantum
+computer has the form
+</p>
+<p>&nbsp;<br>
+$$
+\vert\Psi\rangle = \sum_{i_1i_2\cdots i_N} \Psi_{i_1i_2\cdots i_N} \vert i_1i_2\cdots i_N\rangle.
+$$
+<p>&nbsp;<br>
+</section>
+
+<section>
+<h2 id="sums">Sums </h2>
+
+<p>Here  the sum runs over all qubit values \( i_a \in \{0,1\} \), and
+\( \vert i_1i_2\cdots i_N\rangle \) is a shorthand for the tensor product
+\( \vert i_1i_2\cdots i_N\rangle = \vert i_1\rangle\otimes
+\vert i_2\rangle\otimes\cdots\otimes\vert i_N\rangle \).  The tensor
+\( \Psi_{i_1i_2\cdots i_N} \) can be thought of as a large vector
+containing \( 2^N \) complex values. The potential capabilities of quantum
+computers stem from the fact that this vector is exponentially large
+and, once \( N\gtrsim 50 \), cannot be stored in a classical computer.
+</p>
+</section>
+
+<section>
+<h2 id="quantum-circuits">Quantum circuits </h2>
+
+<p>When one operates a quantum computer, one initializes it with an
+initial state \( \vert \Psi\rangle^{(0)} \) (usually \( \vert \Psi\rangle^{(0)} =
+\vert 000\cdots{}0\rangle \)), and the state of the system evolves according to
+the Schr&#246;dinger equation, which transforms \( \vert \Psi\rangle^{(n)} \) into
+\( \vert \Psi\rangle^{(n+1)} = \hat U^{(n)} \vert \Psi\rangle^{(n)} \), with the
+evolution operator \( \hat U^{(n)} \) given by
+</p>
+<p>&nbsp;<br>
+$$
+\hat U^{(n)} = T e^{-i\int_{t_n}^{t_{n+1}} dt \hat H(t)}.
+$$
+<p>&nbsp;<br>
+
+<p>Here
+where \( T \) is the time-ordering operator and \( \hat H(t) \) is the
+Hamiltonian of the system.
+</p>
+</section>
+
+<section>
+<h2 id="unitary-evolution-operator-in-quantum-computing">Unitary evolution operator in quantum computing </h2>
+
+<p>In physics, the system is described by
+\( \hat H(t) \). In quantum computing, one actually starts with the
+evolution operators \( \hat U^{(n)} \), assuming that someone else has
+worked out how to engineer appropriate Hamiltonians. The different
+evolution operators that one will use are called <b>gates</b> and fall
+into two categories, depending on whether they act on a single qubit
+or on two qubits.
+</p>
+</section>
+
+<section>
+<h2 id="single-qubit-gate">Single-qubit gate </h2>
+<p>A single-qubit gate is defined on one qubit as \( \hat
+U\vert i\rangle = \sum_j U_{ji}\vert j\rangle \), where the \( 2\times 2 \) matrix
+\( U_{ij} \) is unitary. In terms of the wavefunction \( \Psi^{(n)} \), such a
+single-qubit gate on qubit \( a \) translates into a matrix that acts as
+</p>
+
+<p>&nbsp;<br>
+$$
+\Psi_{i_1i_2\cdots{}i_N}^{(n+1)} = \sum_{i'_a} U^{(n)}_{i_ai'_a}
+\Psi_{i_1\cdots{}i_{a-1}i'_a i_{a+1}\cdots{}i_N}^{(n)}.
+$$
+<p>&nbsp;<br>
+</section>
+
+<section>
+<h2 id="two-qubit-gate">Two-qubit gate </h2>
+
+<p>Likewise, a two-qubit gate acting on qubits \( a \) and \( b \) is described
+by a \( 4\times 4 \) matrix and transforms the wavefunction into
+</p>
+<p>&nbsp;<br>
+$$
+\Psi_{i_1i_2\cdots{}i_N}^{(n+1)} = \sum_{i'_a,i'_b} U^{(n)}_{i_ai_b,i'_ai'_b}
+\Psi_{i_1\cdots{}i_{a-1}i'_a i_{a+1}\cdots{}i_{b-1}i'_b i_{b+1}\cdots{}i_N}^{(n)}.
+$$
+<p>&nbsp;<br>
+
+<p>Depending on the quantum hardware, some gates are easier to implement
+than others.
+</p>
+</section>
+
+<section>
+<h2 id="the-pauli-gates">The Pauli gates </h2>
+<p>Typical examples include the one-qubit gates (the first
+three are the Pauli matrices)
+</p>
+
+<p>&nbsp;<br>
+$$
+\begin{eqnarray}
+X &=& \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \\
+Y &=& \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \\
+Z &=& \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \\
+H &=& \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \\
+T &=& \begin{pmatrix} e^{i\pi/8} & 0 \\ 0 & e^{-i\pi/8} \end{pmatrix}.
+\end{eqnarray}
+$$
+<p>&nbsp;<br>
+</section>
+
+<section>
+<h2 id="and-the-controlled-not-or-cnot-two-qubit-gate">And the controlled NOT (or CNOT) two-qubit gate </h2>
+
+<p>&nbsp;<br>
+$$
+C_X = \begin{pmatrix} 1 & 0 & 0 & 0 \\
+                      0 & 1 & 0 & 0 \\
+                      0 & 0 & 0 & 1 \\
+                      0 & 0 & 1 & 0 \end{pmatrix}.
+$$
+<p>&nbsp;<br>
+
+<p>These gates are transformed into so-called quantum circuits (see whiteboard notes) and 
+they are read a little like music, with one line per qubit.
+</p>
+</section>
+
+<section>
+<h2 id="measurement">Measurement </h2>
+<p>The last ingredient of the quantum computer gate model is measurement. When a qubit \( a \) is
+measured, it returns the value \( \alpha \) (\( \alpha = 0 \) or 1) with probability
+</p>
+
+<p>&nbsp;<br>
+$$
+P_\alpha = \sum_{i_1\cdots{}i_{a-1} i_{a+1}\cdots{}i_N} \left|
+\Psi_{i_1\cdots{}i_{a-1} \alpha i_{a+1}\cdots{}i_N}^{(n)} \right|^2.
+$$
+<p>&nbsp;<br>
+</section>
+
+<section>
+<h2 id="after-measurement">After measurement </h2>
+
+<p>The new wavefunction becomes</p>
+
+<p>&nbsp;<br>
+$$
+\Psi_{i_1\cdots{}i_N}^{(n)}\rightarrow
+\delta_{i_a,\alpha} \frac{1}{\sqrt{P_\alpha}}
+\Psi_{i_1\cdots{}i_{a-1} \alpha i_{a+1}\cdots{}i_N}^{(n)}.
+$$
+<p>&nbsp;<br>
+
+<p>And that's essentially it. The above set of equations entirely
+describes what a quantum computer is supposed to do. The entire field
+of quantum algorithms consists of using
+these rules to perform useful computations.
+</p>
+</section>
+
+<section>
+<h2 id="quantum-computing-in-a-nutshell">Quantum computing in a nutshell </h2>
+
+<p>In a nutshell, a quantum computer allows one to perform a subset
+of linear algebra, namely matrix-vector multiplications, with very
+special matrices (the <b>gates</b> that act only on certain indices and
+belong to a fixed set of unitary matrices) on exponentially large
+vectors (the wavefunction \( \Psi_{i_1\cdots{}i_N}^{(n)} \)). The appeal
+of quantum computers clearly comes from the exponentially large size
+of those wavefunctions.
+</p>
+</section>
+
+<section>
+<h2 id="downside">Downside </h2>
+
+<p>However, a very strong downside is that at the
+end of the calculation one does not hold the corresponding
+exponentially large vector of \( 2^N \) values, but only a much smaller
+set: \( N \) bits of (probabilistic) information. We get a single sample
+of the distribution \( |\Psi_{i_1\cdots{}i_N}^{(n)}|^2 \). This is one of
+the two Achilles' heels of quantum computing, which we dub the I/O
+bottleneck (well, more the O bottleneck for this aspect).
+</p>
+</section>
+
+<section>
+<h2 id="inaccesible-parts-of-the-space">Inaccesible parts of the space </h2>
+
+<p>It is very important to realize that the largest part of the Hilbert
+space of dimension \( 2^N \) will remain forever inaccessible to quantum
+computers (and classical methods). This can be understood using a
+simple counting argument. Suppose that the quantum circuit consists of
+\( D \) layers of gates (\( D \) being the depth of the circuit). We also
+suppose that each layer is packed with as many gates as possible
+(meaning that all the qubits are acted upon). Lastly, each gate is
+parametrized by a few angles. Then the total dimension of the subspace
+that can be spanned by these circuits is \( O(D N) \), which is obviously
+much smaller than \( 2^N \).
+</p>
+</section>
+
+<section>
+<h2 id="some-numbers">Some numbers </h2>
+
+<p>Now let us put some realistic numbers. Suppose that we work with
+\( N=100 \) qubits. The total dimension of the Hilbert space is \( 2^{100}
+\approx 10^{30} \). Typical depths that can be considered with existing
+hardware are of the order of \( D\approx 100 \), but let us  suppose that
+this number is scaled up to \( D=10^6 \). The explorable subspace would
+still have 20 orders of magnitude fewer degrees of freedom than the
+full Hilbert space. So the question really is: does this subspace
+belong to the <b>relevant</b> part of the Hilbert space?
+</p>
+</section>
+
+<section>
+<h2 id="degrees-of-freedom">Degrees of freedom  </h2>
+
+<p>And, conversely,
+is the <b>relevant</b> part of the Hilbert space amenable to classical
+simulations? The word relevant is defined very loosely here, but there
+are several scientific articles that start to give it a more precise
+meaning. For instance, the problem of the <b>barren plateaus</b> in the
+variational quantum eigensolver (VQE) algorithm has been traced back
+to the fact that most of the states in the \( O(N D) \)-dimensional
+subspace manifold are essentially chaotic, hence irrelevant.
+</p>
+</section>
+
 <section>
 <h2 id="notations-and-definitions">Notations and definitions  </h2>