You're reading from A Practical Guide to Quantum Machine Learning and Quantum Optimization

Product typeBook

Published inMar 2023

PublisherPackt

ISBN-139781804613832

Edition1st Edition

Concepts

Quantum Computing

Authors (2):

Elías F. Combarro

Samuel González-Castillo

View More author details

1.5 Working with multiple qubits and universality

Now that we have mastered working with two-qubit systems, it will be fairly straightforward to generalize all the notions that we have been studying to the case in which the number of qubits in our circuits is arbitrarily big. You know the drill: we will start by defining, mathematically, what a multi-qubit system is, we will then learn how to measure it and, finally, we will introduce quantum gates that act on many qubits at the same time.

1.5.1 Multi-qubit systems

With all that we have learned so far, it will now be very easy to understand how to work with multi-qubit systems.

As you may have already deduced, if we have qubits, the states that constitute the computational basis are

$\begin{matrix} {\left| 0 \right\rangle \otimes \left| 0 \right\rangle \otimes \cdots \otimes \left| 0 \right\rangle,} & \\ {\left| 0 \right\rangle \otimes \left| 0 \right\rangle \otimes \cdots \otimes \left| 1 \right\rangle,} & \\ {\vdots} & \\ {\left| 1 \right\rangle \otimes \left| 1 \right\rangle \otimes \cdots \otimes \left| 1 \right\rangle.} & \\ \end{matrix}$

We usually omit the $\otimes$ symbol to write

$\begin{matrix} {\left| 0 \right\rangle\left| 0 \right\rangle\cdots\left| 0 \right\rangle,} & \\ {\left| 0 \right\rangle\left| 0 \right\rangle\cdots\left| 1 \right\rangle,} & \\ {\left| 1 \right\rangle\left| 1 \right\rangle\cdots\left| 1 \right\rangle} & \\ \end{matrix}$

$\left| {00\cdots 0} \right\rangle,\left| {00\cdots 1} \right\rangle,\ldots,\left| {11\cdots 1} \right\rangle$

or simply

$\left| 0 \right\rangle,\left| 1 \right\rangle,\ldots,\left| {2^{n} - 1} \right\rangle.$

Important note

When using the $\left| 0 \right\rangle,\left| 1 \right\rangle,\ldots,\left| {2^{n} - 1} \right\rangle$ notation for basis states, the total number of qubits must be clear from context. Otherwise, a state like, for example, $\left| 2 \right\rangle$ might mean either $\left| {10} \right\rangle$ , $\left| {010} \right\rangle$ , $\left| {0010} \right\rangle$ , or any string with leading zeroes and ending in …and that would be an intolerable ambiguity!

Of course, a generic state of the system will then be of the form

$\left| \psi \right\rangle = a_{0}\left| 0 \right\rangle + a_{1}\left| 1 \right\rangle + \ldots + a_{2^{n} - 1}\left| {2^{n} - 1} \right\rangle$

subject to the only condition that the amplitudes $a_{i}$ should be complex numbers such that ${\sum}_{l = 0}^{2^{n} - 1}\left| a_{l} \right|^{2} = 1$ . Our dear old friend, the normalization condition!

To learn more…

Notice that the number of parameters describing the general state of an -qubit system is exponential in . For highly entangled states, we do not know how to represent all this information in a more succinct way and it is strongly suspected that it is not possible. Part of the power of quantum computing comes from this possibility of implicitly working with $2^{n}$ complex numbers by manipulating just qubits.

Exercise 1.18

Check that the basis state $\left| j \right\rangle$ is represented by a $2^{n}$ -dimensional column vector whose -th component is 1, while the rest are 0 (Hint: Use, repeatedly, the expression for the tensor product of column vectors that we discussed in Section 1.4.1 and the fact that the tensor product is associative). Deduce that any -qubit state can be represented by a $2^{n}$ -dimensional column vector with unit length.

If we decide to measure all the qubits of the system in the computational basis, we will obtain with probability $\left| a_{m} \right|^{2}$ . If that is the case, then the state will collapse to $\left| m \right\rangle$ . But if we only measure one of the qubits, say the -th one, then we will obtain with probability

$\sum\limits_{l \in J_{0}}\left| a_{l} \right|^{2},$

where $J_{0}$ is the set of numbers whose -th bit is . In this scenario, the state of the system after measuring would be

$\frac{\sum\limits_{l \in J_{0}}a_{l}\left| l \right\rangle}{\sqrt{\sum\limits_{l \in J_{0}}\left| a_{i} \right|^{2}}}.$

Exercise 1.19

Derive the formulas for the case in which the result of the measurement is .

Exercise 1.20

What is the probability of getting when we measure the second qubit of $\left. (1\slash 2)\left| {100} \right\rangle + (1\slash 2)\left| {010} \right\rangle + \sqrt{\left. 1\slash 2 \right.}\left| {001} \right\rangle \right.$ ? What will the state be after the measurement if we indeed get ?

Computing inner products of -qubit systems in Dirac notation is very similar to doing it with two-qubit systems. The procedure is analogous to the one we showed in Section 1.4.1, but taking into account that

$\left( {\left\langle \psi_{1} \right| \otimes \ldots \otimes \left\langle \psi_{n} \right|} \right)\left( {\left| \varphi_{1} \right\rangle \otimes \ldots \otimes \left| \varphi_{n} \right\rangle} \right) = \left\langle \psi_{1} \middle| \varphi_{1} \right\rangle\ldots\left\langle \psi_{n} \middle| \varphi_{n} \right\rangle.$

Exercise 1.21

Compute the inner product of $\left| x \right\rangle$ and $\left| y \right\rangle$ , where and are both binary strings of length . Use your result to prove that ${\{\left| x \right\rangle\}}_{x \in {\{ 0,1\}}^{n}}$ is, indeed, an orthonormal basis.

Exercise 1.22

Compute the inner product of the states $\sqrt{\left. 1\slash 2 \right.}\left( {\left| {000} \right\rangle + \left| {111} \right\rangle} \right)$ and $\left. 1\slash 2\left( {\left| {000} \right\rangle + \left| {011} \right\rangle + \left| {101} \right\rangle + \left| {110} \right\rangle} \right) \right.$ .

We can now turn to the question of how to operate on many qubits at once. Let's define multi-qubit gates!

1.5.2 Multi-qubit gates

Since -qubit states are represented by $2^{n}$ -dimensional column vectors, -qubit gates can be identified with $2^{n} \times 2^{n}$ unitary matrices. Similar to the two-qubit case, we can construct -qubit gates by taking the tensor product of gates on a smaller number of qubits. Namely, if $U_{1}$ is an $n_{1}$ -qubit gate and $U_{2}$ is an $n_{2}$ -qubit gate, then $U_{1} \otimes U_{2}$ is an $(n_{1} + n_{2})$ -qubit gate and its matrix is given by the tensor product of the matrices $U_{1}$ and $U_{2}$ .

To learn more…

The expression for the tensor product of two matrices and is

\begin{pmatrix} a_{11} & {\ldots} & a_{1q} \\ {\vdots} & \ddots & {\vdots} \\ a_{p1} & {\ldots} & a_{pq} \\ \end{pmatrix} \otimes B = \begin{pmatrix} {a_{11}B} & {\ldots} & {a_{1q}B} \\ {\vdots} & \ddots & {\vdots} \\ {a_{p1}B} & {\ldots} & {a_{pq}B} \\ \end{pmatrix}.

However, there are -qubit gates that cannot be constructed as tensor products of smaller gates. One such example is the Toffoli or CCNOT gate, a three-qubit gate that acts on the computational basis as

$\text{CCNOT}\left| x \right\rangle\left| y \right\rangle\left| z \right\rangle = \left| x \right\rangle\left| y \right\rangle\left| {z \oplus \left( {x \land y} \right)} \right\rangle,$

where $\oplus$ is the XOR function and $\land$ is the symbol for the AND Boolean function. Thus, CCNOT applies a doubly controlled (in this case, by the first two qubits) NOT gate to the third qubit — hence the name!

Exercise 1.23

Obtain the matrix for the CCNOT gate and verify that it is unitary.

The Toffoli gate is important because, using it and with the help of auxiliary qubits, we can construct any classical Boolean operator. For instance, $\text{CCNOT}\left| 1 \right\rangle\left| 1 \right\rangle\left| z \right\rangle = \left| 1 \right\rangle\left| 1 \right\rangle\left| {\neg z} \right\rangle$ (where $\neg z$ is the negation of ) and $\text{CCNOT}\left| x \right\rangle\left| y \right\rangle\left| 0 \right\rangle = \left| x \right\rangle\left| y \right\rangle\left| {x \land y} \right\rangle$ . This shows that, with quantum circuits, we can simulate the behavior of any classical digital circuit at the cost of using some additional ancillary qubits, since any Boolean function can be built with just negations and conjunctions. This is somewhat surprising, because we know that all quantum gates are invertible, while not all Boolean functions are. It then follows that we could make all of our digital circuits reversible just by implementing a classical version of the Toffoli gate!

We will not be studying any other concrete examples of gates that act on three (or more!) qubits because, in fact, we can simulate their behavior with circuits that only use one- and two-qubit gates. Keep on reading to know how!

1.5.3 Universal gates in quantum computing

Current quantum computers can't implement every possible quantum gate. Instead, they rely on universality results that show how any unitary operation can be decomposed as a circuit that uses a reduced set of primitive gates. In previous sections, we mentioned, for instance, that any one-qubit gate can be obtained by using just $R_{Z}$ and $R_{Y}$ rotations. It turns out that similar results exist for the general case of -qubit quantum gates.

To us, it will be important to know that, for any unitary operation, we can construct a circuit that implements it using only one-qubit gates and the CNOT gate. For this reason, we say that those gates are universal — in the same sense that, for example, negation and conjunction are universal for Boolean logic. This fact will be crucial for our study of feature maps and variational forms in connection to quantum neural networks and other quantum machine learning models.

To learn more…

In addition to one-qubit gates plus CNOT, there are many other sets of universal gates. For instance, it can be shown that the three gates , , and CNOT can be used to approximate any unitary operation to any desired accuracy — and they are universal in that sense. See Section 4.5 of the book by Nielsen and Chuang [69] for proofs of these facts and for more examples of universal gate sets.

To illustrate how CNOT and one-qubit gates can be used to implement any other quantum gate, the following circuit shows a possible decomposition of the Toffoli gate targeting the top qubit:

HTTTTHTTT††† .

Exercise 1.24

Verify that the preceding circuit implements the Toffoli gate by checking its action on the states of the computational basis.

This concludes our review of the fundamentals of quantum computing. We've come a long way since the beginning of this chapter, but by now we have mastered all the mathematics that we will need in order to study quantum machine learning and quantum optimization algorithms. Soon, we will see all these concepts in action!

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Authors (2)

Elías F. Combarro

Elías F. Combarro holds degrees from the University of Oviedo (Spain) in both Mathematics (1997, award for second highest grades in the country) and Computer Science (2002, award for highest grades in the country). After some research stays at the Novosibirsk State University (Russia), he obtained a Ph.D. in Mathematics (Oviedo, 2001) with a dissertation on the properties of some computable predicates under the supervision of Prof. Andrey Morozov and Prof. Consuelo Martínez. Since 2009, Elías F. Combarro has been an associate professor at the Computer Science Department of the University of Oviedo. He has published more than 50 research papers in international journals on topics such as Computability Theory, Machine Learning, Fuzzy Measures and Computational Algebra. His current research focuses on the application Quantum Computing to algebraic, optimisation and machine learning problems. From July 2020 to January 2021, he was a Cooperation Associate at CERN openlab. Currently, he is the Spain representative in the Advisory Board of CERN Quantum Technology Initiative, a member of the Advisory Board of SheQuantum and one of the founders of the QSpain, a quantum computing think tank based in Spain.
Read more about Elías F. Combarro

Samuel González-Castillo

Samuel González-Castillo holds degrees from the University of Oviedo (Spain) in both Mathematics and Physics (2021). He is currently a mathematics research student at the National University of Ireland, Maynooth, where he works as a graduate teaching assistant. He completed his physics bachelor thesis under the supervision of Prof. Elías F. Combarro and Prof. Ignacio F. Rúa (University of Oviedo), and Dr. Sofia Vallecorsa (CERN). In it, he worked alongside other researchers from ETH Zürich on the application of Quantum Machine Learning to classification problems in High Energy Physis. In 2021, he was a summer student at CERN developing a benchmarking framework for quantum simulators. He has contributed to several conferences on quantum computing.
Read more about Samuel González-Castillo

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