Unitary operator and analytic solution of time-dependent Schrödinger equation (2) 

Setting a Pauli-X operator as the Hamiltonian and evolving , we get:

In[77]:=

Out[74]=

Return the amplitudes at time :

In[78]:=

Out[78]=

Evolution in a time-dependent field: nuclear magnetic resonance (NMR): (2) 

Set up the Hamiltonian (as a time-dependent operator) in a magnetic field :

In[79]:=

In[80]:=

Out[78]=

Get the state vector for the pure state:

In[81]:=

Out[81]=

PVMs (projective measurements) (5) 

A measurement can be defined by specifying the corresponding measurement basis.

Measure a 3D system in its state basis:

In[82]:=

Out[43]=

In[83]:=

Out[83]=

Test to confirm that the measured states are the same as the basis states:

In[84]:=

Out[84]=

Measure a 2-qubit system in the computational basis:

In[85]:=

Out[86]=

In[87]:=

Out[87]=

Note the labels for the corresponding eigenvalues, from 0 to n-1 as follows:

In[88]:=

Out[88]=

For composite systems, one can measure one or more qudits. This can be done by specifying an order for QuantumMeasurementOperator.

2D×3D composite system:

In[89]:=

Measure only the first qudit:

In[90]:=

Out[90]=

Measure only the 2nd qudit:

In[91]:=

Out[91]=

Measure both qudits:

In[92]:=

Out[92]=

One can use the following format for 'named' bases and their corresponding eigenvalues: QuantumMeasurementOperator[name→eigenvalues].

For example, define a measurement operator with a "named" QuantumBasis (as an eigenbasis) and a list of eigenvalues:

In[93]:=

Out[93]=

POVMs (7) 

One can also give a list of POVM elements by which to define the measurement operator:

In[94]:=

Check that all POVM elements are explicitly positive semidefinite:

In[95]:=

Out[95]=

Check the completeness relations:

In[96]:=

Out[96]=

Measure POVMs on a quantum state:

In[97]:=

Get the post-measurement states:

In[98]:=

Out[98]=

Get the corresponding probabilities:

In[99]:=

Out[99]=

Show post-measurement states are the same as states we defined POVMs initially:

In[100]:=

Out[100]=

Superdense coding (5) 

Alice wants to send Bob two classical bits: 00, 01, 10, or 11. She can do so by using a single qubit if her qubit and Bob's are initially prepared as Bell states:

In[123]:=

Out[124]=

Depending on Alice's intended message, she will perform the following operations on her qubit:

1. To send 00, she does nothing.

2. To send 01, she applies the X-gate.

3. To send 10, she applies the Z-gate.

4. To send 11, she applies the X-gate and then the Z-gate.

Such operations can be represented as circuits, each with a final state resulting from application of its respective gate(s) to the initial Bell state:

In[125]:=

Out[125]=

Next, Alice sends her qubit to Bob through a quantum channel. If Bob performs a Bell measurement on his qubits, he receives Alice's message.

Alice's 'messaging' with Bob can be fully implemented in a quantum circuit using two ancillary qubits. In the circuit, the 1st qubit is Alice's, the 2nd one is Bob's, and the 3rd and the 4th are the ancillary qubits. (Note that Alice sends her qubit to Bob and Bob performs a measurement on qubits 1 and 2):

In[126]:=

Out[114]=

Define an initial state as , where is the code that Alice wants to send Bob, which is encoded in two ancillary qubits (qubits 3 and 4).

Run the state through the circuit and return the outcome probabilities:

In[127]:=

Out[127]=

For each case, Bob finds Alice's code with probability 1.

Quantum teleportation (4) 

Quantum teleportation is the reverse of superdense coding. Here, one wants to teleport a generic unknown quantum bit. Suppose that Alice wants to send a qubit (qubit 1) to Bob. To implement a quantum circuit for teleporting a qubit, Alice and Bob share an entangled state (qubits 2 and 3). Qubits 1 and 2 represent Alice's system, while qubit 3 is Bob's. The goal is to transfer the state of Alice's first qubit to Bob's qubit.

Set up a circuit:

In[128]:=

In[129]:=

Out[129]=

The state to be teleported is =α+β, where α and β are unknown amplitudes. The input state of the circuit is as follows:

In[130]:=

Given the result of Alice's measurement on the first and second qubits, get the post-measurement states:

In[131]:=

Out[114]=

Regardless of measurement results, the state of the third qubit is the same state as the original first qubit =α+β (with only a normalization difference). Trace out the first and second qubits and compare the reduced state of qubit 3 only:

In[132]:=

Out[132]=

Deutsch–Jozsa algorithm (3) 

Alice wants to determine if a Boolean function , used by Bob, is constant for all values of or balanced (where is an -digit binary number between 0 and -1). Consider a balanced implementation of for n=2 as follows: , , , meaning ⊻.

Given n=4, show that XOR is balanced:

In[133]:=

Out[129]=

Consider the case where Alice stores her query in a 2-qubit system, denoted by qubit 1 and qubit 2. Bob will store the answer in the 3rd qubit (the answer qubit). After Bob performs the oracle, Alice measures her qubits. If she gets anything other than , then the oracle is a balanced function.

Implement the algorithm:

In[134]:=

In[135]:=

Out[135]=

Get the action of the quantum circuit as a measurement of the initial state:

In[136]:=

Out[136]=

The measurement indicates that the oracle is a balanced function.

Bernstein–Vazirani algorithm (3) 

The goal of Bernstein–Vazirani algorithm is to find a secret string bit as s using the action of a Bernstein–Vazirani oracle (i.e., which should be treated as a black box) which is defined by this transformation: → with the index register state of n-qubits, and the state of an ancillary qubit carrying the result.

Bernstein–Vazirani oracle for the secret bit of 101:

In[137]:=

Out[138]=

Bernstein–Vazirani circuit for the secret bit of 101:

In[139]:=

Out[139]=

As expected, the only measurement outcome corresponds to the secret bit-string:

In[140]:=

Out[140]=

Grover's search algorithm (14) 

The goal of Grover's search algorithm is to find solutions of a Boolean function . This can be done using the named-circuits or oracles in our quantum framework.

The action of a Boolean oracle is defined by this transformation: → with the index register state of n-qubits, and the state of an ancillary qubit carrying the result of Boolean function f(x); meaning if = then it will be if x is a solution of f(x), and no change if not.

Define a Boolean function of 3-SAT with 5 clauses:

In[141]:=

Truth table of above Boolean function:

In[142]:=

Out[142]=

Corresponding oracle's quantum circuit:

In[143]:=

The diagram of oracle:

In[144]:=

Out[144]=

We shall prepare the 4-qubit in above circuit (ie the ancillary qubit) is 0 state, and then other qubits (1-3) in the index register states. To compare with above truth table, we will create the index register states |x> in the order |2ⁿ-1> down to |0:

In[145]:=

Create a list with elements {|x>,|q⊕f(x)}:

In[146]:=

Out[146]=

The action of a phase oracle can be defined as the following transformation: with the index register state of n-qubits:

In[147]:=

The diagram of oracle:

In[148]:=

Out[148]=

Create a list with elements {|x>,(-1)^f(x)|x>}:

In[149]:=

Out[149]=

Generate the corresponding Grover circuit using a Boolean oracle:

In[150]:=

Out[150]=

Generate a Grover circuit using the phase oracle for a Boolean function:

In[151]:=

Out[151]=

Given a Grover-phase circuit for a Boolean function, calculate the probability of success of the algorithm:

In[152]:=

In[153]:=

Calculate the success probability after each iteration (note 111 is the solution of the Boolean function):

In[154]:=

Plot the success probability:

In[155]:=

Out[155]=

which, as expected, follows the formula for the probability of success where M is number of correct solutions out of total N after k steps

Quantum phase estimation (5) 

The Quantum phase estimation algorithm solves the problem of finding in the eigenvalue equation = where is a unitary operator. The inputs of the algorithm are qubits at the initial state . The output is . We will consider a phase shift as the unitary operator where we aim to find the phase θ:

In[156]:=

Out[156]=

To specify the corresponding quantum circuit, one can use the built-in circuit "PhaseEstimation" that takes two input arguments: a unitary operator U, and an integer n. The integer n specifies the number of qubits and controlled-U^j operators in the circuit, with j=0,1,…,n-1. The accuracy of phase estimation and the success probability depends on n:

In[157]:=

In[158]:=

Out[132]=

Return the corresponding measurement, with all qubits prepared in state:

In[159]:=

In[160]:=

Out[160]=

Given the outcome with the largest probability, estimate the phase:

In[161]:=

Out[161]=

As expected, it is a rough estimate, since we picked a small value for n. If one increases n, with a higher probability, one can get a better estimate of the phase.

Estimate the phase (the expected value is 1/5=0.2) for n=6:

In[162]:=

Out[162]=

Shor's algorithm

Shor's algorithm is an algorithm that can factorize a large number using a quantum computation. Given an integer , Shor's algorithm can return its factors and such that . This task is difficult for a classical computer if is large. The algorithm involves three main steps: classical pre-processing, quantum order-finding, and classical post-processing.

1. Classical preprocessing

- If is even, return 2 and /2.

- If is odd, pick a random number .

- Compute the greatest common divisor (GCD) of a and N. If GCD(, )!=1, return a and /a.

- If GCD(, )= 1, proceed to the order-finding algorithm.

2. Quantum order-finding

- Initialize a 2n qubit register (for ) in the state .

- Apply Hadamard gates to the first qubits.

- Apply modular exponentiation gate , where the subscript indicates the number of qubits.

- Apply the inverse quantum Fourier transform to the first qubits.

- Measure the first qubits in the computational basis.

3. Classical post-processing

- Let s be the integer representation of the measurement outcome. Compute the order r of a^xmod N by finding the solutions to j(/r) = s, where j is an integer.

- If is odd or choose a different a and repeat the process.

- Otherwise, at least one of or is a nontrivial factor of .

The quantum circuit used for quantum order-finding can be schematically illustrated as the following (n =3). (Note that U₄ is the modular exponent gate):

In[163]:=

Out[163]=

Currently, the specific gate implementation of the modular exponentiation for arbitrary and a is unknown. However, there are some specific cases that have been studied. In the following, consider the implementation of quantum order-finding for (, a) = (15, 7) that has been demonstrated in experiments. Although one needs 8 qubits to factor 15 in general, by choosing a specific a, it is possible to use only 7 qubits (3 register, 4 ancilla).

Prepare the quantum order-finding circuit for (, a) = (15, 7):

In[164]:=

In[165]:=

Out[165]=

Initialize the state:

In[166]:=

Apply the Shor circuit to the initial state:

In[167]:=

Out[167]=

In[168]:=

Out[168]=

In decimal notation, the possible states are: , , , and . If one obtains , one must repeat the protocol. Otherwise:

Outcome: |2〉 → j(8/r) = 2 (j, r) = (1, 4)

p=GCD[(7^4/2+1) ,15] = 5 and q= GCD[(7^4/2-1),15] = 3

Outcome: |4〉 → j(8/r) = 4 (j, r) = (1, 2)

p= GCD[(7^2/2+1),15] = 1 and q= GCD[(7^2/2-1),15] = 3

Outcome: |6〉 → j(8/r) = 6 (j, r) = (3, 4)

p=GCD[(7^4/2+1) ,15] = 5 and q= GCD[(7^4/2-1),15] = 3

Thus, with 0.75 probability, one can find at least one nontrivial factor of 15.

Quantum Key Distribution: BB84 scheme (11) 

As an example of a quantum key distribution (QKD), consider the BB84 protocol. Alice generates random bits and for each, randomly chooses either a Z-basis or X-basis of measurement. Consider a sequence of 9 bits:

In[169]:=

Out[169]=

In[170]:=

Out[170]=

Depending on her choice of bit and base, she follows the following protocol for sending a qubit to Bob:

In[171]:=

Out[172]=

With Alice's bits and bases having been generated randomly, she will send Bob these qubits:

In[173]:=

Out[174]=

Bob randomly generates a sequence of either Z-bases or X-bases of measurement:

In[175]:=

Out[175]=

Thus, Bob applies the following measurement operators on the qubits he recieved from Alice:

In[176]:=

Out[176]=

Post-measurement, it is the case that, whenever Bob's measurement basis is not the same as Alice's, he gets two random results:

In[177]:=

Out[177]=

Return one possible realization of Bob' s series of measurements (noting that some results will be consistent and other random, depending on Alice' s qubit and Bob's measurement):

In[178]:=

Out[178]=

Bob assigns the bit 0 whenever he gets or , and the bit 1 for or :

In[179]:=

Out[179]=

Finally, Alice and Bob publicly share with each other the measurement basis that they used (in other words, they conduct a classical communication). If the basis is same, they each keep the bit; if not, they each disregard the corresponding bit. (Note that each does so separately, and each one obtains a secret key, which will be the same as the other's):

In[180]:=

Out[180]=

In[181]:=

Out[181]=

In[182]:=

Out[182]=

Alice and Bob now share a secret key. Note that the key length is less than the number of qubits Alice sent to Bob (because Bob picks the measurement basis randomly, and he might not get the same outcome as Alice). Alice can keep sending Bob qubits until they reach a desired length. This can be implemented as follows:

In[183]:=

For example, the key length is set to 5 bits below:

In[184]:=

Out[184]=