Hadamard gate

Therefore, the one-qubit Hadamard gate [32] can be implemented as a switch that operates within the Hamiltonian Hn swltch and transform its neutral-state component into the negative-state one ... 

Note that a similar Hadamard gate can be realized for the positive-negative switch that operates between the positively and negatively charged sheets of the PES of the Auz-(NH3)2 cluster. A readout of the Auz-(NH3)2 cluster qubits can be achieved in various ways, for instance by measuring its IR spectrum. Each state of the 5 qubits can be detected by IR spectroscopy - the IR identifier of each 5-qubit state is given in Table 3. 

Alice continues her preparation by using another device called the Hadamard gate, which operates on a single photon and does the following ... 

There are also other three important one-qubit gates, which are the phase gate (S), the 3t/8 (T) gate and the Hadamard gate. The operators S and T are defined by the matrices ... 

The Hadamard gate can be decomposed into a linear combination of the X and Z operators ... 

An example of a quantum circuit is illustrated in Figure 3.2. The upper line represents the qubit a) and lower one the qubit b). The operations appearing in the figure mean that the S gate is applied to the first qubit, whereas the gate T is applied to the second one. These operations are followed by the application of a two qubit operation U, and finally by the application of a Hadamard gate to the first qubit only. The whole process can be translated in mathematical language as [H 0 1] C/ [S T] af>). 

The relation above, Equation (3.5.6), is very useful, because it indicates the necessary operations to be performed on each qubit for the implementation of the QFT. For instance, starting from the first qubit, the first step is to apply a Hadamard gate, followed by a relative phase change, controlled by the other qubits, of the system. One can see from Equation (3.5.6) why a relative phase change is needed. This operation can be performed by some applications of the logic gate Rk. ... 

The second operation is the application of the Hadamard gate to the first qubit, ir), leaving the system on state described by ... 

As an example, let us consider the case of two qubits. The input state is = 00), from which an uniform superposition must be created, applying Hadamard gates on each qubit of the system. Therefore the following stage will be ... 

With the eigenket of the operator U at hand, we can use the phase estimation routine for finding the order, r, such as = 1 (mod N). For that, the system must be prepared in the initial state iri i) = 0)/ 1), and a Hadamard gate applied on every t qubit of the first register, in order to create a uniform superposition. The quantum state of the system is then... 

This can be achieved by applying a Hadamard gate to each qubit of the first set, as discussed earlier. 

Another fundamental single-qubit gate is the Hadamard gate. As it can be observed from Equation (4.2.7), a (f) pulse along the y-axis closely resembles a Hadamard matrix H. 

Despite the similarity between the two matrices, a pulse can only be classified as pseudo-Hadamard gate. This is so, because a single ( -) pulse is not self-reversible (that is, when applied twice to a given quantum state it does not recover the original state), which is a fundamental property of a true Hadamard gate. However, for applications where self reversibility is not required, it is common to use a ( -)y pulse in place of a Hadamard gate. To produce a true Hadamard operation in NMR it is necessary to introduce an extra pulse to perform the necessary phase correction in the pulse matrix. This can be achieved by adding a tt-pulse around the z- or x-axis after the ( ) pulse, i.e.. 

The upper indexes, 00), 01), 110), 111), are to emphasize that the corresponding operators only execute a true Hadamard gate when they act on the indicated states. The indexes 01, 12, and 23 indicate the pulse transition as indicated in Figure 4.2. However, the operators c), d) and e), g) can be implemented by a single pulse sequence if we use two-frequency pulses to excite simultaneously two transitions. For example, Uhj. = where 01-23 indicates a two-frequency selective pulse that act simultaneously on the transitions 01 and 23 see Figure 4.2, will implement a operation independently of the initial state. All the Hadamard transformations indicated in (4.2.11) are self-reversible. 

Many algorithms start with an uniform superposition of states, produced by a multi-qubit Hadamard gate. Although such an operation can be formally written as a tensor product of one-qubit Hadamard gate, sometimes we have to regard it as a multi-qubit gate. This is the case, for instance, of quadrupole nuclei. 

Figure 4.14 Representation in the Bloch Sphere of one qubit evolution during the execution of some Ic ic gates, (a) One qubit Hadamard Operation (b) CNOT 10> operation (c) Experimental and simulated evolution of both qubits during the double application of a Hadamard Gate. Ad ted with permission from Reference (30)...

P4.1 - Using the pulse matrix for the transition selective pulses in quadrupolar systems, find the pulse operators for the single-qubit gates NOT a, NOTg, Ha, Hg. Apply such pulse operators to the state 00) and show the they execute the expected actions. Then, show that the two Hadamard gates are self-reversible. 

To obtain the matrix representation of the pulse operators of the Hadamard gates we can use that ... 

Where, regardless a global phase, we see that the operators act as expected. To exemplify the reversibility of the Hadamard gates let us apply them twice to the state 00> ... 

If the first qubit qi is measured and yields qi = 0) = 0 then the second qubit collapses to q2 = ( 0) + 1))- This is not a classical state, but a simple Hadamard gate transforms q2 into a classical state. The Hadamard gate is defined by the matrix... 

It follows that by using the Hadamard gate, there is a clear correlation between the measured values of the first and second qubit. In particular, they always have opposite values. 

A similar scenario can be developed, when the second qubit q2 is measured first. In this case, the first qubit transformed by a Hadamard gate, yields the opposite value of q2. 

If Bob is the one who measures first his qubit qiB, the procedure is simply mirrored. Alice now has to apply a Hadamard gate on her qubit, thus obtaining HqiA- Again Alice and Bob will have measured complementary binary digits. 

To evaluate a specific attack, let us consider that Alice s and Bob s qubits are not really entangled, but Eve has sent qubits of her own choice to both of them. Eve also can listen to the classical channel. The best she can do is send a classical 0 to Alice and a Hadamard 1 to Bob. Actually, all other combinations are equivalent to, or less advantageous than, this one. Alice and Bob decide who is to measure first once they already have the qubits. With 1/2 probabihty, Alice is measuring, in which case the readings are consistent. Bob measures first, again with probability 1/2. Bob will measure a 1 or 0 with equal probability. Then Alice transforms the classical 0 with a Hadamard gate, and also reads a... 

This property is even stronger in our improved algorithm with no classical communication presented in the Conclusion section. Here, the decision on how to measure a qubit (directly or with a Hadamard gate) is not made on a classical communication channel, but in private by Alice and Bob independently. 

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