Chapter 2 - Qubits and Collections of Qubits

Introduction

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 DiVincenzo:2000:

1. Be a scalable physical system with well-defined qubits
2. Be initializable to a simple fiducial state such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 discussed in Chapter 6. This chapter is primarily concerned with the fifth of these criteria. This will enable us to discuss many interesting aspects of quantum information problem while postponing some other technical details regarding the other criteria.

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 that has many available states. These two states are represented by the vectors $\left\vert {0}\right\rangle$ and $\left\vert {1}\right\rangle$ and the qubit could be in the state $\left\vert {0}\right\rangle$, 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

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

This means that this vector is normalized, i.e. its magnitude (or length) is 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)

Thus, the qubit state can be rewritten as

	$\left\vert {\psi }\right\rangle =\left({\begin{array}{c}\alpha _{0}\\\alpha _{1}\end{array}}\right).$	(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 closed system evolution is unitary. (See Appendix C, Sec. C.3.8.) These unitary matrices, or unitary transformations, as well as their generalization to many qubits, transform 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 unitary transformation. Consider a unitary transformation $V$ on a single qubit state $\left\vert {\psi }\right\rangle$. If the result of the transformation is $\left\vert {\psi ^{\prime }}\right\rangle$, we can then 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.

$\left\vert {\psi }\right\rangle$

$\left\vert {\psi ^{\prime }}\right\rangle$

Figure 2.1: Circuit diagram for a one-qubit gate that implements the unitary transformation $V\,\!$. The input state $\left\vert {\psi }\right\rangle$ is on the left and the output, $\left\vert {\psi ^{\prime }}\right\rangle$, is on the right.

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)

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 (refer to Fig. 2.2 below). While this is somewhat confusing, it is important to remember convention; circuit diagrams will become increasingly important as the number of operations grows larger.

$\left\vert {\psi }\right\rangle$

$\left\vert {\psi ^{\prime \prime }}\right\rangle$

Figure 2.2: Circuit diagram for two one-qubit gates that implements the unitary transformation $V\,\!$ followed by another unitary transformation $U\,\!$. Like the single gate, the input state $\left\vert {\psi }\right\rangle$ is on the left and the new output, $\left\vert {\psi ^{\prime \prime }}\right\rangle$, is 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 are fixed. There are several such transformations that 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, and 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , the commutator is defined to be

	$[A,B]=AB-BA.\,\!$	(2.14)

For the two gates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X,\,\!} Eq.(2.7) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y,\,\!} Eq.(2.12) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z \,\!} Eq.(2.9) are called the Pauli matrices. They are also sometimes denoted $\sigma _{x}\,\!$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_y\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_z\,\!} , or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1\,\!} , $\sigma _{2}\,\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_3\,\!} respectively. They are ubiquitous in quantum computing and quantum information processing. This is because they, along with the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \times 2\,\!} identity matrix, form a basis for the set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \times 2\,\!} Hermitian matrices and can be used to describe all $2\times 2$ unitary transformations as well. We will return to the latter point in the next chapter.

TABLE 2.1

Table 2.1: The Pauli Matrices. The table shows the Pauli matrices, three different, but common notations, and the action on a state. The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\,\!} is a binary digit, 0 or 1.

Pauli Matrix	Notation 1	Notation 2	Notation 3	Action
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)\,\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\,\!}	$\sigma _{x}\,\!$	$\sigma _{1}\,\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X|x\rangle = |x\oplus 1\rangle\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)\,\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y =iXZ\,\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_y\,\!}	$\sigma _{2}\,\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y|x\rangle = i(-1)^x|x\oplus 1\rangle\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)\,\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z\,\!}	$\sigma _{z}\,\!$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_3\,\!}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z|x\rangle = (-1)^x|x\rangle\,\!}

To show that they form a basis for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_3\,\!} are arbitrary, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0 + a_3\,\!} and Failed to parse (syntax error): {\displaystyle a_0 − a_3\,\!} are abitrary too. This matrix can be written as

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}A &= a_0 \mathbb{I} + a_1X + a_2Y + 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{align} }	(2.20)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a}\cdot\vec{\sigma} = \sum_{i=1}^3a_i\sigma_i\,\!} is the "dot product" between ${\vec {a}}\,\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i, j, k\,\!} are numbers from the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1, 2, 3\}\,\!} and the definitions for $\delta _{ij}\,\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ijk}\,\!} are given in Eqs. (C.17) and (C.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. 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 $\epsilon _{ijk}=-\epsilon _{jik}\,\!$. This can now be simplified,

	$[\sigma _{i},\sigma _{j}]=2i\epsilon _{ijk}\sigma _{k}.\,\!$	(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 Appendix C, Section C.7.) The set of basis states obtained in this way is

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

This set is more often written in short-hand notation as (again see Appendix C, Section C.7 for details and examples)

	$\left\{\left\vert {00}\right\rangle ,\;\left\vert {01}\right\rangle ,\;\left\vert {10}\right\rangle ,\;\left\vert {11}\right\rangle \right\},\,\!$	(2.26)

which can also be expressed as

	$\left\{\left({\begin{array}{c}1\\0\\0\\0\end{array}}\right),\;\left({\begin{array}{c}0\\1\\0\\0\end{array}}\right),\;\left({\begin{array}{c}0\\0\\1\\0\end{array}}\right),\;\left({\begin{array}{c}0\\0\\0\\1\end{array}}\right)\right\}.\,\!$	(2.27)

The extension to three qubits is straight-forward,

	$\left\{\left\vert {000}\right\rangle ,\;\left\vert {001}\right\rangle ,\;\left\vert {010}\right\rangle ,\;\left\vert {011}\right\rangle ,\;\left\vert {100}\right\rangle ,\;\left\vert {101}\right\rangle ,\;\left\vert {110}\right\rangle ,\;\left\vert {111}\right\rangle \right\}.\,\!$	(2.28)

Those familiar with binary will recognize these as the numbers zero through seven. Thus we consider this an ordered basis. Thus, they can also be acceptably presented as

	$\left\{\left\vert {0}\right\rangle ,\;\left\vert {1}\right\rangle ,\;\left\vert {2}\right\rangle ,\;\left\vert {3}\right\rangle ,\;\left\vert {4}\right\rangle ,\;\left\vert {5}\right\rangle ,\;\left\vert {6}\right\rangle ,\;\left\vert {7}\right\rangle \right\}.\,\!$	(2.29)

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 that may share some information or wish to communicate. In this case, the state belonging to Alice can 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

	$\left\vert {\psi }\right\rangle =\alpha _{00}\left\vert {00}\right\rangle +\alpha _{01}\left\vert {01}\right\rangle +\alpha _{10}\left\vert {10}\right\rangle +\alpha _{11}\left\vert {11}\right\rangle =\left({\begin{array}{c}\alpha _{00}\\\alpha _{01}\\\alpha _{10}\\\alpha _{11}\end{array}}\right).$	(2.30)

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

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

Quantum Gates for Many Qubits

Just as the case for one single qubit, the most general closed-system transformation of a state of many qubits is a unitary transformation. Being able to make an arbitrary unitary transformation on many qubits is an important task. If an arbitrary unitary transformation on a set of qubits can be made, then any quantum gate can be implemented. If this ability to implement any arbitrary quantum gate can be accomplished using a particular set of quantum gates, that set is said to be a universal set of gates or that the condition of universality has been met by this set. It turns out that there is a theorem which provides one way for identifying a universal set of gates.

Theorem:

The ability to implement an entangling gate between any two qubits, plus the ability to implement all single-qubit unitary transformations, will enable universal quantum computing.

It turns out that one doesn’t need to be able to perform an entangling gate between distant qubits; nearest-neighbor interactions are sufficient. We can transfer the state of a qubit to a qubit that is next to the one we would like it to interact with, then perform the entangling gate between the two and then transfer back.

This is an important and often used theorem which will be the main focus of the next few sections. A particular class of two-qubit gates which can be used to entangle qubits will be discussed along with circuit diagrams for many qubits.

Controlled Operations

A controlled operation is one that is conditioned on the state of another part of the system, usually a qubit. The most cited example is the CNOT (controlled NOT) gate, which flips one (target) bit if another qubit is in the state $\left\vert {1}\right\rangle \,\!$; thus it is controlled NOT operation for qubits. This gate is used often enough to warrant detailed discussion here.

Consider the following matrix operation on two qubits:

	$C_{12}=\left({\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{array}}\right).$	(2.31)

Under this transformation, the following changes occur:

	${\begin{array}{c|c}\;\left\vert {\psi }\right\rangle \;&C_{12}\left\vert {\psi }\right\rangle \\\hline \left\vert {00}\right\rangle &\left\vert {00}\right\rangle \\\left\vert {01}\right\rangle &\left\vert {01}\right\rangle \\\left\vert {10}\right\rangle &\left\vert {11}\right\rangle \\\left\vert {11}\right\rangle &\left\vert {10}\right\rangle \end{array}}$	(2.32)

This transformation is called the CNOT, or controlled NOT, since the second bit is flipped if the first is in the state $\left\vert {1}\right\rangle \,\!$ and otherwise left alone. The circuit diagram for this transformation corresponds to the following representation of the gate. Let $x\,\!$ and $y\,\!$ be zero or one. The CNOT is then given by

	$\left\vert {x}\right\rangle _{i}\left\vert {y}\right\rangle _{j}{\overset {CNOT}{\rightarrow }}\left\vert {x}\right\rangle _{i}\left\vert {x\oplus y}\right\rangle _{j}.$	(2.33)

In binary, of course Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\oplus 0 =0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\oplus 1 = 1 = 1\oplus 0} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\oplus 1 =0} . The circuit diagram is given in Fig. 2.3 below. The first qubit at the top of the diagam, $\left\vert {x}\right\rangle$, is called the control bit while the one below, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{y}\right\rangle} , is called the target bit.

Figure 2.3: Circuit diagram for a CNOT gate.

One can immediately generalize the operation of the CNOT to a controlled-U gate. This is a gate, shown in Fig. 2.4, which implements a unitary transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\,\!} on the second qubit, if the state of the first is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{1}\right\rangle\,\!} . The matrix transformation is given by

	$CU_{12}=\left({\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&u_{11}&u_{12}\\0&0&u_{21}&u_{22}\end{array}}\right),$	(2.34)

where the matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \left(\begin{array}{cc} u_{11} & u_{12} \\ u_{21} & u_{22} \end{array}\right).}

For example the controlled-phase gate is given in Fig. 2.5.

Figure 2.4: Circuit diagram for a CU gate.

Many-qubit Circuits

Many qubit circuits are a straight-forward generalization of the single quibit circuit diagrams. For example, Fig. 2.6 shows the implementation of CNOTFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{14}} and CNOTFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{23}} in the same diagram. The crossing of lines is not confusing since there is a target and control which are clearly distinguished in each case.

It is quite interesting however, that as the diagrams become more complicated, the possibility arises that one may change between equivalent forms of a circuit that, in the end,

Figure 2.5: Circuit diagram for a Controlled-phase $C_{PHASE}\!$ gate.

Figure 2.6: Multiple CNOT gates on a set of qubits.

implements the same multiple-qubit unitary. For example, noting that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(C_{PHASE})H = CNOT\,\!} , the two circuits in Fig. 2.7 implement the same two-qubit unitary transformation. This enables the simplication of some quite complicated circuits.

Figure 2.7: Two circuits which are equivalent since they implement the same two-qubit unitary transformation.

Measurement

Measurement in quantum mechanics is quite different from that of classical mechanics. In classical mechanics (and computing), one assumes that a measurement can be made at will without disturbing or changing the state of the physical system. In quantum mechanics, this assumption cannot be made. This is important for a variety of reasons that will become clear later.

Standard Prescription

In the introduction a simple example was provided to distinguish quantum states from classical states. This example of two wells with one particle can (with caution) be used here as well.

Consider the quantum state in a superposition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert 0\right\rangle\,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert 1\right\rangle\,\!} of the form

	$\left\vert \psi \right\rangle =\alpha _{0}\left\vert 0\right\rangle +\alpha _{1}\left\vert 1\right\rangle ,\,\!$	(2.35)

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha_0|^2 + |\alpha_1|^2 = 1\,\!} . If the state is measured in the computational basis, the result will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert 0\right\rangle\,\!} with probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha_0|^2\,\!} and $\left\vert 1\right\rangle \,\!$ with probability $|\alpha _{1}|^{2}\,\!$. As always, it is important to note that it is not in either of the computational bases but a superposition of the two.

This can be easily shown by acting on the state $\left\vert \psi \right\rangle \,\!$ with a Hadamard transformation,

	$H\left\vert \psi \right\rangle =\left\vert 0\right\rangle .\,\!$	(2.36)

This state, produced from a unitary transformation of $\left\vert \psi \right\rangle \,\!$, has probability $0\,\!$ of being in the state $\left\vert 1\right\rangle \,\!$ and probability $1\,\!$ of being in the state $\left\vert 0\right\rangle \,\!$. If it were in one or the other, then acting on the state with a Hadamard transformation would give some probability of it being in $\left\vert 0\right\rangle \,\!$ and some probability of being in $\left\vert 1\right\rangle \,\!$. (This argument is so simple and pointed that it was taken almost word-for-word from Mermin's book, page 27.)

A measurement in the computational basis is said to project this state into either the state $\left\vert 0\right\rangle \,\!$ or the state $\left\vert 1\right\rangle \,\!$ with probabilities $|\alpha _{0}|^{2}\,\!$ and $|\alpha _{1}|^{2}\,\!$ respectively. To understand this as a projection, consider the following way in which the $0\,\!$ -component of the state $\left\vert \psi \right\rangle \,\!$ is found. The state $\left\vert \psi \right\rangle \,\!$ is projected onto the the state $\left\vert 0\right\rangle \,\!$ mathematically by taking the inner product (see Section C.4) of $\left\vert 0\right\rangle \,\!$ and $\left\vert \psi \right\rangle \,\!$

	$\left\langle 0\mid \psi \right\rangle =\alpha _{0}.\,\!$	(2.37)

Notice that this is a complex number and that its complex conjugate can be expressed as

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\psi \mid 0\right\rangle = \alpha_0^*. \,\!}	(2.38)

Therefore the probability can be expressed as

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle\psi\mid 0 \right\rangle \left\langle 0\mid\psi\right\rangle = \left\vert\left\langle 0\mid \psi\right\rangle \right\vert^2.\,\!}	(2.39)

Now consider a multiple-qubit system with state

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert \Psi\right\rangle = \sum_i \alpha_i\left\vert i\right\rangle.\,\!}	(2.40)

The result of a measurement is a projection and the state is projected onto the basis state $\left\vert i\right\rangle \,\!$ with probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\alpha_i|^2\,\!} ---the same properties are true of this more general system.

To summarize, if a measurement is made on the system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert\Psi\right\rangle\,\!} , the result Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert i\right\rangle\,\!} is obtained with probability $|\alpha _{i}|^{2}\,\!$. Assuming that $\left\vert i\right\rangle \,\!$ results from the measurement, the state of the system has been projected into the state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert i\right\rangle\,\!} . Therefore, the state of the system immediately after the measurement is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert i\right\rangle\,\!} .

A circuit diagram with a measurement represented by a box with an arrow is given in Figure 2.8.

Figure 2.8: The circuit diagram for a measurement.

An alternative is to put an "M" inside the box. This is shown in Fig. 2.9.

Figure 2.9: An alternative circuit diagram for a measurement.

As an example, the measurement result can be used for input for another state. The unitary transform in Figure 2.10 is one that depends upon the outcome of the measurement. Notice that the information input, since it is classical, is represented by a double line.

Figure 2.10: A circuit which includes a measurement.

Projection Operators

Projection operators are used quite often and the description of measurement in the previous section is a good example of how they are used. One may ask, what is a projector? In ordinary three-dimensional space, a vector is written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v=v_x\hat{x}+v_y\hat{y}+v_z\hat{z}\,\!} and the $x\,\!$ part of the vector can be obtained by

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{x}(\hat{x}\cdot\vec v) = v_x\hat{x}.\,\!}	(2.40)

This is the part of the vector lying along the x axis. Notice that if the projection is performed again, the same result is obtained

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{x}(\hat{x} \cdot v_x\hat{x}) = v_x\hat{x}.\,\!}	(2.41)

This is (the) characteristic of projection operations. When one is performed twice, the second result is the same as the first.

This can be extended to the complex vectors in quantum mechanics. The outer product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{x}\right\rangle\!\!\left\langle{x}\right\vert\,\!} is a projector. For example, $\left\vert {0}\right\rangle \!\!\left\langle {0}\right\vert \,\!$ is a projector and can be written in matrix form as

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{0}\right\rangle\!\!\left\langle{0}\right\vert = \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right). \,\!}	(2.42)

Acting with this on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{\psi}\right\rangle = \alpha_0\left\vert{0}\right\rangle + \alpha_1\left\vert{1}\right\rangle\,\!} gives

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right) \left(\begin{array}{c} \alpha_0 \\ \alpha_1 \end{array}\right) = \left(\begin{array}{c} \alpha_0 \\ 0 \end{array}\right). \,\!}	(2.43)

Acting again produces

	$\left({\begin{array}{cc}1&0\\0&0\end{array}}\right)\left({\begin{array}{c}\alpha _{0}\\0\end{array}}\right)=\left({\begin{array}{c}\alpha _{0}\\0\end{array}}\right).\,\!$	(2.44)

This is due to the fact that

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\left\vert{0}\right\rangle\!\!\left\langle{0}\right\vert)^2 = \left\vert{0}\right\rangle\!\!\left\langle{0}\right\vert. \,\!}	(2.45)

In fact, this property essentially defines a projection. A projection is a linear transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P\,\!} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^2 = P\,\!} . Much of our intuition about geometric projections in three-dimensions carries to the more abstract cases. One important example is that the sum over all projections is the identity. The generalization to arbitrary dimensions, where $\left\vert {i}\right\rangle \,\!$ is any basis vector in that space, is immediate. In this case the identity, expressed as a sum over all projectors, is

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i} \left\vert{i}\right\rangle\!\!\left\langle{i}\right\vert = 1. \,\!}	(2.46)

Phase in/Phase out

The probability of finding the system in the state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{x}\right\rangle\,\!} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0\,\!} or $1\,\!$, is

	${\begin{aligned}{\mbox{Prob}}_{\left\vert {\psi }\right\rangle }(\left\vert {x}\right\rangle )&=\left\langle {\psi }\mid {x}\right\rangle \left\langle {x}\mid {\psi }\right\rangle \\&=|\left\langle {\psi }\mid {x}\right\rangle |^{2}.\end{aligned}}$	(2.47)

Note that $\left\vert {\psi }\right\rangle \,\!$ and $\left\langle {\psi }\right\vert \,\!$ both appear in this expression. So if $\left\vert {\psi ^{\prime }}\right\rangle =e^{-i\theta }\left\vert {\psi }\right\rangle \,\!$ were substituted into the expression for ${\mbox{Prob}}(\left\vert {x}\right\rangle )\,\!$, then the expression is unchanged,

	${\begin{aligned}{\mbox{Prob}}_{\left\vert {\psi ^{\prime }}\right\rangle }(\left\vert {x}\right\rangle )&=\left\langle {\psi ^{\prime }}\mid {x}\right\rangle \left\langle {x}\mid {\psi ^{\prime }}\right\rangle \\&=e^{-i\theta }\left\langle {\psi }\mid {x}\right\rangle \left\langle {x}\mid {\psi }\right\rangle e^{i\theta }\\&=|\left\langle {\psi }\mid {x}\right\rangle |^{2}.\end{aligned}}$	(2.48)

Therefore when $\left\vert {\psi }\right\rangle \,\!$ changes by a phase, there is no effect on this probability. This is why it is often said that

	$\left({\begin{array}{cc}e^{i\theta }&0\\0&e^{-i\theta }\end{array}}\right)=e^{i\theta }\left({\begin{array}{cc}1&0\\0&e^{-i2\theta }\end{array}}\right)\,\!$	(2.49)

is equivalent to

	$\left({\begin{array}{cc}1&0\\0&e^{-2i\theta }\end{array}}\right).\,\!$	(2.50)

However, there are times when a phase can make a difference. In those cases it is really a relative phase between two states that makes the difference. This will become clear later on.

Continue to Chapter 3 - Physics of Quantum Information