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Therefore the square of the operator will be a diagonal matrix with all eigenvalues squared. The kinetic energy operator will then be given by different eigenvalues in x and y spatial dimensions. We can now evaluate the unitary operator in it’s matrix form in order to construct quantum gates which can be later implemented in the quantum circuit. According to Eq. (4) the position operator is a diagonal matrix.

Figure 5 Therefore the square of the operator will be a diagonal matrix with all eigenvalues squared. The kinetic energy operator will then be given by different eigenvalues in x and y spatial dimensions. We can now evaluate the unitary operator in it’s matrix form in order to construct quantum gates which can be later implemented in the quantum circuit. According to Eq. (4) the position operator is a diagonal matrix.

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All position operators act multiplicatively in Cartesian co-ordinate space. The operator for (x“)? can also be calculated directly by multiplying x@ with itself. We can calculate the momentum operator in a similar manner. However, the momentum operators do not act multiplica- tively. Also we have already expanded our total wave- function in an eigenspectrum of position eigenfunctions. Calculating the momentum eigenvalues for each position eigenfunction can be a laborious task for large systems. A much more efficient approach needs to be taken to make a discrete quantum Fourier transform of the wave func- tion. This takes the wave function to momentum space where the momentum operators act multiplicatively and the momentum eigenvalues will be the same as the po- sition eigenvalues for respective discretized space points. An inverse discrete quantum Fourier transform can be done to bring back the function into Cartesian-space. The momentum operator can then be applied as a N x N matrix given by FIG. 1. Discrete two dimensional space. The colour bar indi- cates the amount of stretch required to map these points to harmonic oscillator potential.
All position operators act multiplicatively in Cartesian co-ordinate space. The operator for (x“)? can also be calculated directly by multiplying x@ with itself. We can calculate the momentum operator in a similar manner. However, the momentum operators do not act multiplica- tively. Also we have already expanded our total wave- function in an eigenspectrum of position eigenfunctions. Calculating the momentum eigenvalues for each position eigenfunction can be a laborious task for large systems. A much more efficient approach needs to be taken to make a discrete quantum Fourier transform of the wave func- tion. This takes the wave function to momentum space where the momentum operators act multiplicatively and the momentum eigenvalues will be the same as the po- sition eigenvalues for respective discretized space points. An inverse discrete quantum Fourier transform can be done to bring back the function into Cartesian-space. The momentum operator can then be applied as a N x N matrix given by FIG. 1. Discrete two dimensional space. The colour bar indi- cates the amount of stretch required to map these points to harmonic oscillator potential.
A remarkable observation reveals each diagonal ele- ment of U(t) makes an exact Taylor series expansion of an exponential function with symmetry about the [8,8] element which will be unity
A remarkable observation reveals each diagonal ele- ment of U(t) makes an exact Taylor series expansion of an exponential function with symmetry about the [8,8] element which will be unity
FIG. 2. Mapping the mesh of discrete space over that har- monic oscillator potential. This gives us an intuitive picture of the potential energy of the particle at particular space points.
FIG. 2. Mapping the mesh of discrete space over that har- monic oscillator potential. This gives us an intuitive picture of the potential energy of the particle at particular space points.
The benefit of this approach lies in the first element being unitary. The first basis of our four qubit system 0000) is incapable of picking up any phase term inadver- tently. The rest of the diagonal terms will provide phase to the next 15 basis states in the sequence. The unitary operator for the kinetic energy can be formulated in a similar manner by making subsequent quantum Fourier transforms as explained in Eq. (5). The complete unitary transformation can be given by We can now implement these different phases in a quantum circuit independently. Taking the [1,1] element a common multiple which will act as a global phase, we can write the unitary operator in the final form
The benefit of this approach lies in the first element being unitary. The first basis of our four qubit system 0000) is incapable of picking up any phase term inadver- tently. The rest of the diagonal terms will provide phase to the next 15 basis states in the sequence. The unitary operator for the kinetic energy can be formulated in a similar manner by making subsequent quantum Fourier transforms as explained in Eq. (5). The complete unitary transformation can be given by We can now implement these different phases in a quantum circuit independently. Taking the [1,1] element a common multiple which will act as a global phase, we can write the unitary operator in the final form
FIG. 3. An efficient quantum circuit for implementing the quantum Fourier transform. where the phase of individual rotation gates can be cal- culated using
FIG. 3. An efficient quantum circuit for implementing the quantum Fourier transform. where the phase of individual rotation gates can be cal- culated using
FIG. 4. The mirror image of Q.F.T circuit which conjugate phases to implement the inverse quantum Fourier transform.
FIG. 4. The mirror image of Q.F.T circuit which conjugate phases to implement the inverse quantum Fourier transform.
We can take out the first element as a global phase. This will give us the first element as unity. FIG. 5. A filter for the |0001) state which detects the state of all 4 qubits then performs a rotation on the 4°” qubit. The series of Toffolli gates after cU1 reverses all changes done to the initial state.
We can take out the first element as a global phase. This will give us the first element as unity. FIG. 5. A filter for the |0001) state which detects the state of all 4 qubits then performs a rotation on the 4°” qubit. The series of Toffolli gates after cU1 reverses all changes done to the initial state.
FIG. 6. Quantum circuit for implementing Us(t) for a 2-qubit system.
FIG. 6. Quantum circuit for implementing Us(t) for a 2-qubit system.
FIG. 7. A generalized quantum circuit for quantum Fourier transform of n-qubit system. will be a diagonal matrix with 2” position eigen values each lying on the diagonal. The exponential expansion of the unitary operator gives a Taylor exponential series in each diagonal element. Inductively, each element will have a constant phase difference and therefore the sys- tem with n-parameters can be expressed using a single parameter. Hence, without any approximation we can create [x“] for a n-qubit system. The quantum circuit for n qubits can be implemented using the same algorithm proposed above; by placing control gates and anti-contro gates on |1) and |0) respectively. Then sending the state information of n*” qubit to the (n—1)*” ancilla in order to trigger the desired phase rotation gate on the required qubit only. This phase rotation will happen only for a particular state of n-qubits based on our filter configu- rations. A series of mirrored Toffolli gates can be used to undo all changes done by the filter. The qubits then can proceed to the next filter afresh. A series of (n-1) such filters are required to execute U. ae Another series of (n-1) such filters when squeezed between the Q.F.T and inverse Q.F.T gates can be used to execute ul,
FIG. 7. A generalized quantum circuit for quantum Fourier transform of n-qubit system. will be a diagonal matrix with 2” position eigen values each lying on the diagonal. The exponential expansion of the unitary operator gives a Taylor exponential series in each diagonal element. Inductively, each element will have a constant phase difference and therefore the sys- tem with n-parameters can be expressed using a single parameter. Hence, without any approximation we can create [x“] for a n-qubit system. The quantum circuit for n qubits can be implemented using the same algorithm proposed above; by placing control gates and anti-contro gates on |1) and |0) respectively. Then sending the state information of n*” qubit to the (n—1)*” ancilla in order to trigger the desired phase rotation gate on the required qubit only. This phase rotation will happen only for a particular state of n-qubits based on our filter configu- rations. A series of mirrored Toffolli gates can be used to undo all changes done by the filter. The qubits then can proceed to the next filter afresh. A series of (n-1) such filters are required to execute U. ae Another series of (n-1) such filters when squeezed between the Q.F.T and inverse Q.F.T gates can be used to execute ul,
FIG. 8. A representation of cross multiplication of proba- bility amplitudes at t = z, where T is total time period of oscillation.
FIG. 8. A representation of cross multiplication of proba- bility amplitudes at t = z, where T is total time period of oscillation.
FIG. 9. Probability amplitudes over the first half of the Oscillation cycle for a 2-qubit simuation.
FIG. 9. Probability amplitudes over the first half of the Oscillation cycle for a 2-qubit simuation.
FIG. 10. Probability amplitudes over the second half of the Oscillation cycle for a 2-qubit simulation.
FIG. 10. Probability amplitudes over the second half of the Oscillation cycle for a 2-qubit simulation.
FIG. 11. Probability amplitudes over the second half of the Oscillation cycle for a 4-qubit simualtion.
FIG. 11. Probability amplitudes over the second half of the Oscillation cycle for a 4-qubit simualtion.
FIG. 12. Probability amplitudes over the second half of the Oscillation cycle for a 4-qubit simulation.
FIG. 12. Probability amplitudes over the second half of the Oscillation cycle for a 4-qubit simulation.
We ignore the sy factor for reducing mathematical clutter and write the individual elements of the Unitary matrix in the table below The Unitary operator will then be evaluated as Clearly, each element is a Taylor exponential expansion and the unitary matrix can be written as,
We ignore the sy factor for reducing mathematical clutter and write the individual elements of the Unitary matrix in the table below The Unitary operator will then be evaluated as Clearly, each element is a Taylor exponential expansion and the unitary matrix can be written as,
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Quantum Simulation
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