Glossary

Quantum computation is marked by a set of unitary matrices (quantum gates) acting on qubit states followed by measurement. The most used representation is the circuit model of computation, comprising straight lines and boxes. The horizontal lines represent qubits and boxes represent single qubit unitary gates. A two qubit unitary gate links one qubit from another via vertical lines. Some useful notations are given below.

Quantum States

• $|+\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle ),\quad |-\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle )$
  
• Bell/ EPR pairs:
• GHZ States:
• W States:

Unitary Operations

• X (NOT gate): $X|0\rangle \,\to \,\ |1\rangle ,\quad X|1\rangle \,\to \,\ |0\rangle ,\quad X|+\rangle \,\to \,\ |+\rangle ,\quad *X|-\rangle \,\to \,\ -|-\rangle$
• Z (Phase gate): $Z|+\rangle \,\to \,\ |-\rangle ,\quad Z|-\rangle \,\to \,\ |+\rangle ,\quad Z|0\rangle \,\to \,\ |0\rangle ,\quad Z|1\rangle \,\to \,\ -|1\rangle$
  

Thus, $|0\rangle$, $|1\rangle$ are eigenstates of Z gate and $|+\rangle$, $|-\rangle$ are eigenstates of X gate.

• H (Hadamard gate): $H|0\rangle \,\to \,\ |+\rangle$ or $H|1\rangle \,\to \,\ |-\rangle$

$X=\left[{\begin{array}{cc}0&1\\1&0\\\end{array}}\right],\quad Z=\left[{\begin{array}{cc}1&0\\0&-1\\\end{array}}\right],\quad H={\frac {1}{\sqrt {2}}}\left[{\begin{array}{cc}1&1\\1&-1\\\end{array}}\right]\quad$

• Controlled-U(CU): uses two inputs, control qubit and target qubit. It operates U on the second(target) qubit only when the first (source) qubit is 1. C-U gates are used to produce entangled states, when the target qubit is $|+\rangle$ and control qubit is not an eigenstate of U. In the given equation 'i' denotes the source qubit and 'j', the target qubit. Following are two important C-U gates.

${\text{Controlled-NOT(CX or CNOT): }}CX_{ij}|+\rangle _{i}|0\rangle _{j}\,\to \,\ {\frac {1}{\sqrt {2}}}(|0_{i}0_{j}\rangle +|1_{i}1_{j}\rangle )$
${\text{Controlled-Phase(CZ): }}CZ_{ij}|+\rangle _{i}|+\rangle _{j}\,\to \,\ {\frac {1}{\sqrt {2}}}(|0_{i}+_{j}\rangle +|1_{i}-_{j}\rangle )$

The commutation relations for the above gates are as follows:
$XH=HZ,\quad XZ=-ZX$
$(X\otimes I)CZ=CZ(X\otimes Z),\quad (Z\otimes I)CZ=CZ(Z\otimes I)$

Hierarchy of Quantum Gates

There are different class of quantum gates as follows,

• Pauli Gates(U): Single Qubit Gates I (Identity), X, Y, Z. All the gates in this set follow $U^{2}=I$
• Clifford Gates(C): Pauli Gates, Phase Gate, C-NOT. This set of gates can be simulated on classical computer. All the gates in this set follow CU=U'C, where U and U' are two different Pauli gates depending on C
• Universal Set of gates: This set consists of all Clifford gates and one Non-Clifford gate (T gate). If a model can realise Universal Set of gates, it can imlement any quantum computation efficiently. T gates follow $UT=P^{a}U'T$, where P is the phase gate and U, U' are any two Pauli gates depending on C. Parameter $a\epsilon {0,1}$ is obtained from U, such that $P^{0}=I$, $P^{1}=P$.
  

To summarize, if $C^{1}=$P, $C^{2}=$C, $C^{3}=$T, then $C^{k}=\{U:UQU=C^{k-1}|Q\epsilon C^{1}\}$

Magic States

It is any state that, if you have an unlimited supply of them, can be used to give you universal quantum computation when used in conjunction with perfect Clifford operations. The standard example is that if you can produce the state $(|0\rangle +e^{i\pi /4}|1\rangle )/{\sqrt {2}}$, then you can combine this with Clifford operations in order to apply a T gate, and we know that T+Clifford is universal.

Universal Resource

A set of $|+_{\theta }\rangle$states on which applying Clifford operations is enough for universal quantum computation.

Gate Teleportation

The idea comes from one-qubit teleporation. This means that one can transfer an unknown qubit |ψi without actually sending it via a quantum channel. The underlying equations explain the notion. See Figure 1 for circuit.

Figure 1: One Bit Teleportation

$(H\otimes I)(CZ_{12})|\psi \rangle _{1}|+\rangle _{2}$

$(H\otimes I)(CZ_{12})(a|0\rangle _{1}+b|1\rangle _{1})|+\rangle _{2}$ $(H\otimes I)(a|0\rangle _{1}|+\rangle _{2}+b|1\rangle _{1}|-\rangle _{2})$ $a|+\rangle _{1}|+\rangle _{2}+b|-\rangle _{1}|-\rangle _{2}$ $|0\rangle _{1}\otimes (a|+\rangle _{2}+b|-\rangle _{2})+|1\rangle _{1}\otimes (a|+\rangle _{2}-b|-\rangle _{2})$ $|0\rangle _{1}\otimes (a|+\rangle _{2}+b|-\rangle _{2})+|1\rangle _{1}\otimes X(a|+\rangle _{2}+b|-\rangle _{2})$ $|0\rangle _{1}\otimes H(a\rangle 0\rangle _{2}+b\rangle 1\rangle _{2})+|1\rangle _{1}\otimes XH(a|0\rangle _{2}+b|1\rangle _{2})$ $|0\rangle _{1}\otimes H|\psi \rangle _{2}+|1\rangle _{1}\otimes X|\psi \rangle _{2}$ $|m\rangle \otimes X^{m}H|\psi \rangle$

Similarly if we have the input state rotated by a $\mathrm {Z} (\theta )$ gate the circuit would look like Figure 2a. As the rotation gate $\mathrm {Z} (\theta )$ commutes with Controlled-Phase gate. Hence, Figure 2b is justified.

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  2(a)Modified Input
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  2(b)Gate Teleportation

This shows that for a pair of $\mathrm {CZ}$ entangled qubits, if the second qubit is in $|+\rangle$ state (not an eigen value of $\mathrm {Z}$) then one can teleport (transfer) the first qubit state operated by any unitary gate $\mathrm {U}$ to the second qubit by performing operations only on the first qubit and measuring it. Next, we would need to make certain Pauli corrections (in this case ${\mathrm {X} }^{\mathrm {m} }$) to obtain $\mathrm {U} |\psi \rangle$. In other words, we can say the operated state is teleported to the second qubit by a rotated basis measurement of the first qubit with additional Pauli corrections.

Graph states

The above operation can also be viewed as a graph state with two nodes and one edge. The qubit 1 is measured in a rotated basis $\mathrm {HZ} (\theta )$, thus leaving qubit 2 in desired state and Pauli Correction $\mathrm {X} ^{|}\mathrm {s1} \rangle \mathrm {HZ} (\theta _{1})|\psi \rangle$, where $\mathrm {s1}$ is the measurement outcome of qubit 1. See Figure 3

Figure 3: Graph State for Single Qubit Gates

Now, suppose we need to operate the state with two unitary gates $\mathrm {Z} (\theta _{1})$ and $\mathrm {Z} (\theta _{2})$. This can be done by taking the output state of $\mathrm {Z} (\theta _{1})$ gate as the input state of $\mathrm {Z} (\theta _{2})$ gate and then repeating gate teleportation for this setup, as described above. Thus, following the same pattern for graph states we have now three nodes (two measurement qubits for two operators and one output qubit) with two edges, entangled as one dimensional chain (See Figure 4).

Figure 4: Gate Teleporation for Multiple Single Qubit Gates

The measurement on qubit $\mathrm {1}$ will operate $\mathrm {X} ^{\mathrm {s1} }\mathrm {HZ} (\theta _{1})|\psi \rangle \otimes I$ on qubits $\mathrm {2}$ and $\mathrm {3}$. If qubit $\mathrm {2}$ when measured in the given basis yields outcome $\mathrm {s2}$, qubit $\mathrm {3}$ results in the following state ${\mathrm {X} }^{\mathrm {s2} }\mathrm {HZ} (\theta _{2}){\mathrm {X} }^{\mathrm {s1} }\mathrm {HZ} (\theta _{1})|\psi \rangle$. Using the relation we shift all the Pauli corrections to one end i.e. qubit $\mathrm {3}$ becomes ${\mathrm {X} }^{\mathrm {s2} }\mathrm {HZ} (\theta _{2})\mathrm {HZ} (\theta _{1})|\psi \rangle$ (${\mathrm {Z} }^{\mathrm {s1} }\mathrm {H} ={\mathrm {HX} }^{\mathrm {s1} }$). This method of computation requires sequential measurement of states i.e. all the states should not be measured simultaneously. As outcome of qubit $\mathrm {1}$ can be used to choose sign of $\pm \theta _{2}$. This technique is also known as adaptive measurement. With each measurement, the qubits before the one measured at present have been destroyed by measurement. It is a feed-forward mechanism, hence known as one way quantum computation.

Measurement Based Quantum Computation (MBQC)

MBQC is a formalism used for quantum computation by operating only single qubit measurements on a fixed set of entangled states, also known as graph states. Graph states denote any graph where each node represents a quantum state, and the edges denote entanglement between any two vertices. The measurement on successive layers of qubits is decided by previous measurement outcomes. Outcomes of last qubit layer gives the result of concerned computation. Following, we illustrate certain primitives necessary to understand the working of MBQC.

Cluster States

In case of multi-qubit quatum circuits, one needs a 2-dimensional graph state. Cluster State is a square lattice used as substrate for such computation. All the nodes are in $|+\rangle$ entangled by $\mathrm {CZ}$ indicated by the edges. It is known to be universal i.e. it can simulate any quatum gate.

Figure 5: Cluster State

Each row would thus represent the teleporation of starting qubit in that row horizontally. On the other hand, vertical edges indicate different input qubits linked with multi-qubit gates (same as circuit model). For example, see Figure Figure 6 to understand the conversion from circuit model to graph state model. As the computation relation follows $X=HZH$, thus, Figure Figure 6a represents Circuit diagram for $\mathrm {CNOT}$ gate in terms of $\mathrm {CZ}$ gate and Single Qubit Gate $\mathrm {H}$.

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  (a)Circuit Diagram to implement C-NOT
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  (b)Graph State Pattern for C-NOT
Figure 6: Measurement Pattern from Circuit Model

Figure 6b shows implementation of the first Hadamard gate on the second input state as measurement $\mathrm {M} _{\mathrm {2} }$ on qubit $\mathrm {2}$. Then $\mathrm {CZ}$ gate is implemented by the entangled qubits $\mathrm {3}$ and $\mathrm {1}$ in the graph state. Qubit $\mathrm {3}$ is entangled to another qubit $\mathrm {4}$ to record the output while measurement $\mathrm {M} _{\mathrm {3} }$ on qubit $\mathrm {3}$ implements the second Hadamard gate. Finally, the states to which qubits $\mathrm {(1)}$ and $\mathrm {(4)}$ are reduced to determine the output states of the two input qubits after $\mathrm {CNOT}$ gate operation.
It is evident that one needs to remove certain nodes from the cluster state in order to implement the above shown graph state. This can be done by Z-basis measurements. Such measurements would leave the remaining qubits in the cluster state with extra Pauli corrections. This can be explained as follows. Consider a two-dimesional graph state $\{\mathrm {1,2} \}$. If qubit $\mathrm {1}$ is to be eliminated, we operate $\mathrm {CZ}$ with $\mathrm {2}$ as target and $\mathrm {1}$ as control.

${\mathrm {CZ} }_{\mathrm {12} }{|+\rangle }_{1}|+\rangle _{2}=|0\rangle _{1}|+\rangle _{2}+|1\rangle _{1}|-\rangle _{2}$

Thus, if measurement on $\mathrm {1}$ yields $\mathrm {m}$, qubit $\mathrm {2}$ would be in the state $\mathrm {Z} ^{\mathrm {m} }|+\rangle$. Hence, such Z-basis measurements invoke an extra $\mathrm {Z} ^{\mathrm {m} }$ Pauli correction on all the neighbouring sites of 1 with 1 eliminated, in the resulting graph state. Thus, to summarise, we design a measurement pattern from gate teleportation circuit of the desired computation as shown above. The cluster state is converted into the required graph state by Z-basis measurement on extraneous sites. Measuring all the qubits in the required basis and we get the required computation in the form of classical outcome register from measurement of the last layer of qubits. If it is a quantum function, the last layer of qubits is the output quantum register.

Brickwork States

Although cluster states are universal for MBQC, yet we need to tailor these to the specific computation by performing some computational (Z) basis measurements. If we were to use this principle for blind quantum computing, Client would have to reveal information about the structure of the underlying graph state. Thus, for the UBQC protocol, we introduce a new family of states called the Brickwork states which are universal for X − Y plane measurements and thus do not require the initial computational basis measurements. It was later shown that the Z-basis measurements can be dropped for cluster states and hence cluster states are also universal in X-Y measurements.

Figure 7: Brickwork State

Definition 1 A brickwork state Gn×m, where m ≡ 5 (mod 8), is an entangled state of n × m qubits constructed as follows (see also Figure 7):

1. Prepare all qubits in state |+i and assign to each qubit an index (i,j), i being a column (i ∈ [n]) and j being a row (j ∈ [m]).
2. For each row, apply the operator c-Z on qubits (i,j) and (i,j + 1) where 1 ≤ j ≤ m − 1.
3. For each column j ≡ 3 (mod 8) and each odd row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2).
4. For each column j ≡ 7 (mod 8) and each even row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2).