PVM (projective measurements) (6) 

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

Measurement of a one-qudit system in the state basis:

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Post-measurement states per measurement result:

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Measurement of a two-qudit system in the computational basis:

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One can get the post-measurements states of measurement results:

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For composite systems, one can specify measuring of one or more qudits. It can be done by specifying an order for QuantumMeasurementOperator. For example, measurement of a only 1st qudit of a two-qudit system in the state basis:

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Check the post-measurement states are as and

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One can use the following format for named basis and specifying corresponding eigenvalues: QuantumMeasurementOperator[name->eigenvalues]. For example, define measurement operator by a QuantumBasis object (as an eigenbasis) and a list of eigenvalues:

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The representation of quantum state and operator can be changed according to any basis transformation. Basis transformation is applicable to QuantumState, QuantumOperator, and QuantumMeasurementOperator as demonstrated below:

Basis change for a quantum state:

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Basis change for a quantum operator:

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POVM (3) 

One can also give a list of POVM elements to define measurement operator.

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Check that all POVM elements are explicitly positive semi-definite:

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Check the completeness relations:

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Post-measurement states after measuring POVMs:

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Quantum Teleportation (4) 

We shall implement a quantum circuit for teleporting a qubit. The 1st and 2nd qubits represent Alice’s system, while the last one is Bob’s. The double lines carry classical bits:

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

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Let’s look at post-measurement states, given the result of Alice’s measurement on 1st and 2nd qubits:

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As one can see, regardless of measurement results, the state of 3rd qubit is the same state as the original 1st qubit =α+β (with only a normalization difference). Tracing out 1st and 2nd qubit, and compare the reduced state of only qubit 3:

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Deutsch-Jozsa algorithm (2) 

The main goal is to determine if a Boolean function f(x) is constant for all values of x or balanced, where x is a n-digit binary number between 0 and 2ⁿ-1, for example when n=2 we have state. We shall consider a balanced implementation of f(x) for n=2 as follows: f(0,0)=0, f(0,1)=f(1,0)=1, f(1,1)=0.

Implement the algorithm:

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Action of quantum circuit on the initial state :

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As one can see, measuring first two qubits give non-zero 11, indicating a balanced function.

Bernstein-Vazirani Algorithm (1) 

For oracle, given a state we have the following transformation:→(-1)^s.x, with s being a secret string

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an example, lets take the secret string as the 4th state of a 3-qubit system ()

Define the circuit as a list of operators: Hadamards -> Oracle -> Hadamards

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Let’s go through a specific example for 2 qubits (ie n=2) and a secret string 11 (ie the 4th state s=4, meaning )

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As one can see, the secret string was and we got it as the final state.

Let's go through another example for 3 qubits and a secret string

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As one can see, the section string was the 5th state () and we got it as the final state

Grover's Search Algorithm (5) 

One can describe the Grover’s algorithm as the search for one marked entry which is an n-bit binary number. As the oracle, one can define the following unitary operator:

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As an example, define oracle for marking state :

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Check the only element of the operator matrix which is -1 is the same as the number of corresponding binary number.

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We shall implement 3-qubit Grover algorithm.

Define Grover circuit using a marker state:

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Define initial state:

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Return the final state (output of Grover circuit on the initial state)

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As one can see, the max probability is for the 5th state, .

Now let’s search for another entry, for example .

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Search for another entry, for example .

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Quantum Phase Estimation (7) 

The Quantum Phase Estimation (QPE) algorithm solves the problem of finding θ in the eigenvalue equation U|ψ〉 = e^2πiθ|ψ〉 where U is a unitary operator. The input and output of this algorithm are as follows:

INPUT: n+1 qubits at the initial state|ψ〉⊗|0〉^⊗n

OUTPUT: |ψ〉⊗|2ⁿθ〉

Here, we will use QPE to estimate the value of π. This is possible if we choose , which is QantumOperator[{"P",1}], since U|1〉=eⁱ|1〉 and so QPE will return |1〉⊗|2^n-1/π〉. Thus, by measuring the final state, we can estimate the value of π. Below, we will provide a step-by-step example of the algorithm for n = 3.

Construct the QPE circuit:

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Initialize the input state

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Feed the input state to the circuit and extract π from the measurement statistic:

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The distribution peaks at the state 001 which is equivalent to integer 1. So, we have 2^3-1/π≃1 or π ≃ 4.

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Below is the code to build PiEstimationCircuit for any n:

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Let's compute π with 6 qubits (n =5):

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Shor's Algorithm

Shor's algorithm is an algorithm that can factorize a big number using a quantum computer. Given an integer N, Shor's algorithm can return its factors p, q such that N = p × q. This task is difficult for a classical computer if N is large. The algorithm can be summarized in three main steps: classical pre-processing, quantum order-finding, and classical post-processing. Quantum computer is used only on the second step.

1. Classical preprocessing

- If N is even, return 2 and N/2.

- If N is odd, pick a random number 1<a<N

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

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

2. Quantum order-finding

- Initialize 2n (for N = 2ⁿ-1) qubit register in state |0〉^⊗(2n-1)⊗|1〉.

- Apply Hadamard gates H^⊗n to the first n qubit.

- Apply modular exponentiation gate U|x〉_n|y〉_n=|x〉_n|y⊕a^xmod N〉_n where n subscript indicate the number of qubits.

- Apply inverse Quantum Fourier Transform to the first n qubits.

- Measure the first n qubits in 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(2ⁿ/r) = s where j is an integer.

- If r is odd or a^r/2=-1mod N choose a different a and repeat the process.

- Otherwise, at least one of gcd(a^r/2+1, N) or gcd(a^r/2-1, N) is a non-trivial factor of N.

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:

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Currently, the specific gates implementation of the modular exponentiation for arbitrary N and a is unknown. However, there are some specific cases that have been studied. Below, we will consider the implementation of quantum order-finding for (N, a) = (15, 7) that has been demonstrated in experiment. Although we need 8 qubits to factor 15 in general, by choosing specific a, it is possible to use only 7 qubits (3 register, 4 ancilla) as demonstrated below.

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Initialize the qubits state :

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In decimal notation, the possible states above are: |000〉, |010〉, |100〉, and |110〉. If we obtain |000〉, we need to 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, we can find at least one non-trivial factor of 15.

Superdense Coding (3) 

In quantum teleportation we send a single qubit by sending two classical bits over a classical channel. Superdense coding is the reverse process. We send two classical bits using a single qubit and a quantum channel. First, Alice and Bob must sharesa maximally entangled state: where subscripts A and B mean the qubit is held by Alice and Bob respectively. Then, depending on Alice intended message, she performs the following operations to her qubit:

1. If Alice wants to send 00, she does nothing to her qubit.

2. If Alice wants to send 01, she performs X-gate to her qubit.

3. If Alice wants to send 10, she performs Z-gate to her qubit.

4. If Alice wants to send 11, she performs X-gate and then Z-gate to her qubit.

After that, Alice sends her qubit to Bob through a quantum channel. Finally, if now Bob performs a Bell measurement on his qubits, he is guaranteed to receive Alice's message as demonstrated below.

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Define a initial state as , with xy the code Alice wants to send Bob:

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As one can see, for any , Bob will get the same state with 100% probability:

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Quantum Cryptography (13) 

Cryptography is the study of secure communications. One of the promises of quantum technology is to provide an inherently secure way to send information. The study of exploiting quantum mechanical properties for cryptographic tasks is known as quantum cryptography. One of the well-studied cases of quantum cryptographic protocols is secure quantum key distribution. Before we can understand this protocol, we need to understand a classical communication protocol known as one-time-pad.

In the one-time-pad scenario, Alice wants to send a secret bit string to Bob. To do this securely, they share a common bit key. Alice encodes her message by performing binary addition between her message and the key. She then sends the encoded message through a public channel to Bob. Bob can decode her message by performing the same binary addition. An eavesdropper can not know Alice's message because he does not have access to the key.

Define Alice's message:

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Define the shared key:

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Alice encodes her message by binary addition:

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Bob decodes the message by binary addition:

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So, using one-time-pad protocol, Alice and Bob can securely communicate. However, we have assumed that no one else has access to the bit key other than Alice and Bob. How can we generate a shared key between Alice and Bob securely? This is the problem of key distribution that can be solved by a quantum BB84 protocol as follows:

1. Alice generates a random bit key and a random basis sequence between Pauli-X and Pauli-Z bases. 2. If the basis is Pauli-Z, she encodes the corresponding bit key in the , basis. If the basis is Pauli-X, she encodes the corresponding bit key in the basis. For example, if her bit keys are 0100 and the bases are XZZX then Alice’s encoding qubits are . 3. Alice sends these qubits to Bob through a quantum channel. 4. Bob also generates a random sequence of basis between Pauli-X and Pauli-Z bases. He then measures Alice's qubits with respect to that basis sequence and keeps the measurement outcome private. 5. Alice and Bob both announce their choice of bases. Then they discard the bits in which their corresponding bases do not match. The remaining bits of Alice and Bob are guaranteed to be perfectly correlated and thus Alice and Bob have shared a key.

Alice generates a bit key and a random basis sequence of length 5:

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Alice encodes her bit key into a 10-qubits state:

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Bob Also generate a random basis sequence:

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Bob measures Alice's qubits in the chosen basis:

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Let's suppose Bob obtains the following outcome:

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After Bob obtains his result, Alice and Bob announce their basis choice publicly and decide which bits are to be discarded:

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Alice's final key:

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Bob's final key:

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In this particular implementation, Alice and Bob successfully share a 4-bits key. They can repeat the procedure until the key has the same length as the length of the intended message. This protocol is secure because prior to the bases choice's public announcement, an eavesdropper can only extract the keys by guessing the measurement basis. However, by doing this he risks disturbing the state. So, to detect an eavesdropper, Alice and Bob can sacrifice some of their key bits to make sure that their bits perfectly correlate. If they do not correlate, this means Alice and Bob have detected the presence of an eavesdropper.