Chapter 2 - Qubits and Collections of Qubits

Qubits and Collections of Qubits

There are several parts to any quantum information processing task. Some of these were written down and discussed by David DiVincenzo in the early days of quantum computing research and are therefore called DiVincenzo’s requirements for quantum computing. These include, but are not limited to, the following, which will be discussed in this chapter. Other requirements will be discussed later.

Five requirements [3]:

1. Be a scalable physical system with well-defined qubits
2. Be initializable to a simple fiducial state such as $\left\vert {000...}\right\rangle$
3. Have much longer decoherence times than gating times
4. Have a universal set of quantum gates
5. Permit qubit-specific measurements

The first requirement is a set of two-state quantum systems which can serve as qubits. The second is to be able to initialize the set of qubits to some reference state. In this chapter, these will be taken for granted. The third concerns noise and noise has become known by the term decoherence. The term decoherence has had a more precise definition in the past, but here it will usually be synonymous with noise. Noise and decoherence will be the topics of later sections. The fourth and fifth will be discussed in this chapter.

Backwards is it? Not from a computer science perspective or from a motivational perspective. Besides, to a large extent, the first two rely very heavily on experimental physics and engineering. These topics are primarily beyond the scope of this introductory material, but will be treated superficially in Chapter 6.

Qubit States

As mentioned in the introduction, a qubit, or quantum bit, is represented by a two-state quantum system. It is referred to as a two-state quantum system, although there are many physical examples of qubits which are represented by two different states of a quantum system which has many available states. These two states are represented by the vectors $\left\vert {0}\right\rangle$ and $\left\vert {1}\right\rangle$ and qubit could be in the state $\left\vert {0}\right\rangle$, or the state $\left\vert {1}\right\rangle$, or a complex superposition of these two. A qubit state which is an arbitrary superposition is written      as

a

$\left\vert {\psi }\right\rangle =\alpha _{0}\left\vert {0}\right\rangle +\alpha _{1}\left\vert {1}\right\rangle ,$

(2.1)

where $\alpha _{0}$ and $\alpha _{1}$ are complex numbers. Our objective is to use these two states to store and manipulate information. If the state of the system is confined to one state, the other, or a superposition of the two, then

$|\alpha _{0}|^{2}+|\alpha _{1}|^{2}=1.\,\!$

(2.2)

Thus this vector is normalized, i.e. it has magnitude, or length one. The set of all such vectors forms a two-dimensional complex (so four-dimensional real) vector space. The basis vectors for such a space are the two vectors $\left\vert {0}\right\rangle$ and $\left\vert {1}\right\rangle$ which are called computational basis states. These two basis states are represented by

$\left\vert {0}\right\rangle =\left({\begin{array}{c}1\\0\end{array}}\right),\;\;\left\vert {1}\right\rangle =\left({\begin{array}{c}0\\1\end{array}}\right).$

(2.3)

Therefore,

$\left\vert {\psi }\right\rangle =\left({\begin{array}{c}\alpha _{0}\\\alpha _{1}\end{array}}\right).$

(2.4)

Qubit Gates

During a computation, one qubit state will need to be taken to a different one. In fact, any valid state should be able to be operated upon to obtain any other state. Since this is a complex vector with magnitude one, the matrix transformation required for closedsystem evolution is unitary. (See Appendix D, Sec. D.2.) These unitary matrices, or unitary transformations, as well as their generalization to many qubits, transform a one complex vector into another and are also called quantum gates, or gating operations. Mathematically, we may think of them as rotations of the complex vector and in some cases (but not all) correspond to actual rotations of the physical system.

Circuit Diagrams for Qubit Gates

Unitary transformations are represented in a circuit diagram with a box around the untary transformation. Consider a unitary transformation $V$ on a single qubit state $\left\vert {\psi }\right\rangle$. If the result of the transformation is $\left\vert {\psi ^{\prime }}\right\rangle$ then we write

$\left\vert {\psi ^{\prime }}\right\rangle =V\left\vert {\psi }\right\rangle .$

(2.5)

The corresponding circuit diagram is shown in Fig. 2.1.

Notice that the diagram is read from left to right. This means that if two consecutive gates are implemented, say $V$ first and then $U$, the equation reads

$\left\vert {\psi ^{\prime \prime }}\right\rangle =UV\left\vert {\psi }\right\rangle .$

(2.6)

(enter figure)

However, the circuit diagram will have the boxes in the reverse order from the equation, i.e. $V$ on the left and $U$ on the right.

Examples of Important Qubit Gates

There are, of course, an infinite number of possible unitary transformations that we could implement on a single qubit since the set of unitary transformations can be parameterized by three parameters. However, a single gate will contain a single unitary transformation, which means that all three parameters a fixed. There are several such transformations which are used repeatedly. For this reason, they are listed here along with their actions on a generic state $\left\vert {\psi }\right\rangle =\alpha _{0}\left\vert {0}\right\rangle +\alpha _{1}\left\vert {1}\right\rangle$. Note that one could also completely define the transformation by its action on a complete set of basis states.

The following is called an “x” gate, or a bit-flip,

$X=\left({\begin{array}{cc}0&1\\1&0\end{array}}\right).$

(2.7)

Its action on a state $\left\vert {\psi }\right\rangle$ is to exchange the basis states,

$X\left\vert {\psi }\right\rangle =\alpha _{0}\left\vert {1}\right\rangle +\alpha _{1}\left\vert {0}\right\rangle ,$

(2.8)

for this reason it is also sometimes called a NOT gate. However, this term will be avoided because a general NOT gate does not exist for all quantum states. (It does work for all qubit states, but this is a special case.)

The next gate is called a phase gate or a “z” gate. It is also sometimes called a phase-flip, and is given by

$Z=\left({\begin{array}{cc}1&0\\0&-1\end{array}}\right).$

(2.9)

The action of this gate is to introduce a sign change on the state $\left\vert {1}\right\rangle$ which can be seen through

$Z\left\vert {\psi }\right\rangle =\alpha _{0}\left\vert {0}\right\rangle -\alpha _{1}\left\vert {1}\right\rangle ,$

(2.10)

The term phase gate is also used for the more general transformation

$P=\left({\begin{array}{cc}e^{i\theta }&0\\0&e^{-i\theta }\end{array}}\right).$

(2.11)

For this reason, the z-gate will either be called a “z-gate” or a phase-flip gate.

Another gate closely related to these, is the “y” gate. This gate is

$Y=\left({\begin{array}{cc}0&-i\\i&0\end{array}}\right).$

(2.12)

The action of this gate on a state is

$Y\left\vert {\psi }\right\rangle =-i\alpha _{1}\left\vert {0}\right\rangle +i\alpha _{0}\left\vert {1}\right\rangle =-i(\alpha _{1}\left\vert {0}\right\rangle -\alpha _{0}\left\vert {1}\right\rangle )$

(2.13)

From this last expression, it is clear that, up to an overall factor of Failed to parse (syntax error): {\displaystyle −i} , this gate is the same as acting on a state with both $X$ and $Z$ gates. However, the order matters. Therefore, it should be noted that

$XZ=-iY,\,\!$

whereas

$ZX=iY.\,\!$

The fact that the order matters should not be a surprise to anyone since matrices in general do not commute. However, such a condition arises so often in quantum mechanics, that the difference between these two is given an expression and a name. The difference between the two is called the commutator and is denoted with a $[\cdot ,\cdot ]$. That is, for any two matrices, $A$ and $B$, the commutator is defined to be

$[A,B]=AB-BA.\,\!$

(2.14)

For the two gates $X$ and $Z$,

$[X,Z]=-2iY.\,\!$

(2.15)

A very important gate which is used in many quantum information processing protocols, including quantum algorithms, is called the Hadamard gate,

$H={\frac {1}{\sqrt {2}}}\left({\begin{array}{cc}1&1\\1&-1\end{array}}\right).$

(2.16)

In this case, its helpful to look at what this gate does to the two basis states:

$H\left\vert {0}\right\rangle ={\frac {1}{\sqrt {2}}}(\left\vert {0}\right\rangle +\left\vert {1}\right\rangle ),$
$H\left\vert {1}\right\rangle ={\frac {1}{\sqrt {2}}}(\left\vert {0}\right\rangle -\left\vert {1}\right\rangle ).$	(2.17)

So the Hadamard gate will take either one of the basis states and produce an equal superposition of the two basis states. This is the reason it is so-often used in quantum information processing tasks. On a generic state

$H\left\vert {\psi }\right\rangle =[(\alpha _{0}+\alpha _{1})\left\vert {0}\right\rangle +(\alpha _{0}-\alpha _{1})\left\vert {1}\right\rangle ].$

(2.18)

The Pauli Matrices

The three matrices $X,Y,$ and $Z$ are called the Pauli matrices. They are also sometimes denoted $\sigma _{x}\,\!$, $\sigma _{y}\,\!$ and $\sigma _{z}\,\!$, or $\sigma _{1}\,\!$, $\sigma _{2}\,\!$ and $\sigma _{3}\,\!$ respectively. They are ubiquitous in quantum computing and quantum information processing. This is because they, along with the $2\times 2\,\!$ identity matrix, form a basis for the set of $2\times 2\,\!$ Hermitian matrices and can be used to describe all $2\times 2$ unitary transformations as well. We will return to this latter point in the next chapter.

To show that they form a basis for $2\times 2$ Hermitian matrices, note that any such matrix can be written in the form

$A=\left({\begin{array}{cc}a_{0}+a_{3}&a_{1}+ia_{2}\\a_{1}-ia_{2}&a_{0}-a_{3}\end{array}}\right).$

(2.19)

Since $a_{0}\,\!$ and $a_{3}\,\!$ are arbitrary, $a_{0}+a_{3}\,\!$ and Failed to parse (syntax error): {\displaystyle a_0 − a_3\,\!} are abitrary too. This matrix can be written as

${\begin{aligned}A&=a_{0}\mathbb {I} +a_{1}X+a_{2}Y+a_{3}Z\\&=a_{0}\mathbb {I} +a_{1}\sigma _{1}+a_{2}\sigma _{2}+a_{3}\sigma _{3}\\&=a_{0}\mathbb {I} +{\vec {a}}\cdot {\vec {\sigma }},\\\end{aligned}}$

(2.20)

where ${\vec {a}}\cdot {\vec {\sigma }}=\sum _{i=1}^{3}a_{i}\sigma _{i}\,\!$ is the "dot product" beteen ${\vec {a}}\,\!$ and ${\vec {\sigma }}=(\sigma _{1},\sigma _{2},\sigma _{3})\,\!$.

An important and useful relationship between these is the following (which shows why the latter notation above is so useful)

$\sigma _{i}\sigma _{j}=\mathbb {I} \delta _{ij}+i\epsilon _{ijk}\sigma _{k},$

(2.21)

where $i,j,k\,\!$ are numbers from the set $\{1,2,3\}\,\!$ and the defintions for $\delta _{ij}\,\!$ and $\epsilon _{ijk}\,\!$ are given in Eqs. (D.17) and (D.8) respectively. The three matrices $\sigma _{1},\sigma _{2},\sigma _{3}\,\!$ are traceless Hermitian matrices and they can be seen to be orthogonal using the so-called Hilbert-Schmidt inner product which is defined, for matrices$A\,\!$ and $B\,\!$, as

$(A,B)={\mbox{Tr}}(A^{\dagger }B).$

(2.22)

The orthogonality for the set is then summarized as

$(\sigma _{i},\sigma _{j})={\mbox{Tr}}(\sigma _{i}\sigma _{j})=2\delta _{ij}.\,\!$

(2.23)

This property is contained in Eq. (2.21). This one equation also contains all of the commutators. By subtracting the equation with the product reversed

$[\sigma _{i},\sigma _{j}]=(\mathbb {I} \delta _{ij}+i\epsilon _{ijk}\sigma _{k})-(\mathbb {I} \delta _{ji}+i\epsilon _{jik}\sigma _{k}),$

(2.24)

but $\delta _{ij}=\delta _{ji}\,\!$ and $\epsilon _{ijk}=-\epsilon _{jik}\,\!$ so

$[\sigma _{i},\sigma _{j}]=2i\epsilon _{ijk}\sigma _{k}.\,\!$

(2.25)

States of Many Qubits

Let us now consider the states of several (or many) qubits. For one qubit, there are two possible basis states, say $\left\vert {0}\right\rangle \,\!$ and $\left\vert {1}\right\rangle \,\!$. If there are two qubits, each with these basis states, basis states for the two together are found by using the tensor product. (See Section D.6.) The set of basis states obtained in this way is

$\left\{\left\vert {0}\right\rangle \otimes \left\vert {0}\right\rangle ,\;\left\vert {0}\right\rangle \otimes \left\vert {1}\right\rangle ,\;\left\vert {1}\right\rangle \otimes \left\vert {0}\right\rangle ,\;\left\vert {1}\right\rangle \otimes \left\vert {1}\right\rangle \right\}.\,\!$

(2.26)

This set is more often written as

$\left\{\left\vert {00}\right\rangle ,\;\left\vert {01}\right\rangle ,\;\left\vert {10}\right\rangle ,\;\left\vert {11}\right\rangle \right\},\,\!$

(2.27)

which can also be expressed as

$\left\{\left({\begin{array}{c}1\\0\\0\\0\end{array}}\right),\;\left({\begin{array}{c}0\\1\\0\\0\end{array}}\right),\;\left({\begin{array}{c}0\\0\\1\\0\end{array}}\right),\;\left({\begin{array}{c}0\\0\\0\\1\end{array}}\right)\right\}.\,\!$

(2.28)

The extension to three qubits is straight-forward

$\left\{\left\vert {000}\right\rangle ,\;\left\vert {001}\right\rangle ,\;\left\vert {010}\right\rangle ,\;\left\vert {011}\right\rangle ,\;\left\vert {100}\right\rangle ,\;\left\vert {101}\right\rangle ,\;\left\vert {110}\right\rangle ,\;\left\vert {111}\right\rangle \right\}.\,\!$

(2.29)

Those familiar with binary will recognize these as the numbers zero through seven. Thus we consider this an ordered basis with the following notation also perfectly acceptable

$\left\{\left\vert {0}\right\rangle ,\;\left\vert {1}\right\rangle ,\;\left\vert {2}\right\rangle ,\;\left\vert {3}\right\rangle ,\;\left\vert {4}\right\rangle ,\;\left\vert {5}\right\rangle ,\;\left\vert {6}\right\rangle ,\;\left\vert {7}\right\rangle \right\}.\,\!$

(2.30)

The ordering of the products is important because each spot corresponds to a physical particle or physical system. When some confusion may arise, we may also label the ket with a subscript to denote the particle or position. For example, two different people, Alice and Bob, can be used to represent distant parties which may share some information or may wish to communicate. In this case, the state belonging to Alice may be denoted $\left\vert {\psi }\right\rangle _{A}\,\!$. Or if she is referred to as party 1 or particle 1, $\left\vert {\psi }\right\rangle _{1}\,\!$.

The most general 2-qubit state is written as

$\left\vert {\psi }\right\rangle =\alpha _{00}\left\vert {00}\right\rangle +\alpha _{01}\left\vert {01}\right\rangle +\alpha _{10}\left\vert {10}\right\rangle +\alpha _{11}\left\vert {11}\right\rangle =\left({\begin{array}{c}\alpha _{00}\\\alpha _{01}\\\alpha _{10}\\\alpha _{11}\end{array}}\right).$

(2.31)

The normalization condition is $|\alpha _{00}|^{2}+|\alpha _{01}|^{2}+|\alpha _{10}|^{2}+|\alpha _{11}|^{2}=1.\,\!$ The generalization to an arbitrary number of qubits, say $n$, is also rather straight-forward and can be written as

$\left\vert {\psi }\right\rangle =\sum _{i=0}^{2^{n}-1}\alpha _{i}\left\vert {i}\right\rangle .\,\!$