Dense coding is a protocol introduced by Bennett and Wiesner [24] that allows two bits of classical information to be transmitted between two partners (Alice and Bob) that share an EPR pair by performing local operations on just a single qubit (Alice’s) of the entangled pair, which is then eventually sent to Bob.

Dense coding can be implemented on the IBM 5Q using the circuit shown in Figure 7, designed by Mermin [25] . The two uppermost horizontal wires in the circuit in Figure 7 represent the two classical bits that are to be sent by Alice (i.e. | x i 〉 = { | 0 〉 , | 1 〉 } ); this two bit string is generated using X gates (i.e. | x 1 x 0 〉 = X x 1 ⊗ X x 0   | 00 〉 ). The two lowermost horizontal wires represent the qubits shared by Alice and Bob. The initial Hadamard gate and cNOT generate the entangled pair in the state

| ϕ + 〉 = 1 2   ( | 00 〉 + | 11 〉 ) .

Then, the second cNOT and the controlled-Z implement the transformations of the protocol. Finally, the third cNOT and the last Hadamard gate transform the resulting Bell state into one of the four computational basis states before a joint measurement in this basis is carried out by Bob in order to obtain the information sent over by Alice.

The results obtained using the ibmqx 4 are shown in Table 1. The protocol is successfully completed around 83%, 74%, 78% and 75% of the times for the sequences 00, 01, 10 and 11, respectively.

Certainly, the complete protocol should be carried out in different locations and include the physical transmission of one of the entangled qubits in order to be of practical use, but a proof of principle as the one shown here seems to be a necessary previous step before the experimentally much more complicated full protocol can take place (a previous step that, as the results show, it is not

completed successfully with sufficiently high accuracy yet). It can be argued that superconducting qubits are not the ideal system to be sent over large distances, but then it should be considered how dense coding is going to be incorporated into a quantum computer that functions with superconducting qubits (if it is going to be so at all).

The quantum Fourier transform (QFT) is a basic unitary transformation in the field of Quantum Computation. For a given state of the computational basis | j 〉 ≡ | j n − 1 ⋅ ⋅ ⋅ j 1 j 0 〉 , with j i = { 0 , 1 } , it is defined (in its product representation) as:

| j n − 1 ⋅ ⋅ ⋅ j 1 j 0 〉 → 1 2 n ( | 0 〉 + e 2 π i   0. j 0 | 1 〉 ) ( | 0 〉 + e 2 π i   0. j 1 j 0 | 1 〉 ) ⋅ ⋅ ⋅   ( | 0 〉 + e 2 π i   0. j n − 1 j n − 2 ⋯ j 0 | 1 〉 ) , (1)

where 0. j n − 1 j n − 2 ⋯ j 0 = j n − 1 / 2 + j n − 2 / 2 2 + ⋯ + j 0 / 2 n represents a binary fraction. An efficient way of implementing the QFT for states of three qubits is shown in Figure 8. To implement the circuit shown in Figure 8 on the IBM 5Q, we use the identity in Figure 9, where R z ( λ ) is a rotation of λ radians around the Z axis on the Bloch sphere:

R z ( λ ) ≡ ( 1 0 0 e i λ ) . (2)

The performance of the QFT of different computational-basis states can be assessed using a simple procedure that takes into account the fact that the state of each qubit in the product state (1) lies on the equator of the Bloch sphere. Hence, applying R z ( λ ) gates to all qubits (with an appropriate value of λ for

each qubit), the state of every qubit can be taken to the closest of { | + 〉 , | − 〉 } on the sphere. In this way, finally measuring in the X basis should result in a single three-bit string with probability equal to 1 in the ideal case.

For the 3-qubit case, the initial states chosen to be Fourier transformed are | 000 〉 and | 011 〉 . For the state | 000 〉 , the QFT―Equation (1)―gives the state:

1 2 2 ( | 0 〉 + | 1 〉 ) ( | 0 〉 + | 1 〉 ) ( | 0 〉 + | 1 〉 ) , (3)

while the QFT of the state | 011 〉 gives:

1 2 2 ( | 0 〉 − | 1 〉 ) ( | 0 〉 + e 3 π i / 2 | 1 〉 ) ( | 0 〉 + e 3 π i / 4 | 1 〉 ) (4)

The results obtained for the two states (3) and (4) on the ibmqx 4 are shown in Table 2. Since, as explained above, the expected results are 000 and 101 with probability equal to 1 for the QFT of the states | 000 〉 and | 011 〉 respectively, it can be said that these states have been Fourier transformed with reliabilities of around 74.8 ± 1.1 % and 61.5 ± 1.0 % (the errors are the standard deviations among the 5 runs). QFTs of states of more than three qubits were not tried out because the limited connectivity among qubits (shown in Figure 1) would result in low-performance circuits.

Bell’s theorem states that no deterministic local hidden-variable theory can reproduce all predictions of Quantum Mechanics [26] . Bell derived an inequality for the singlet-spin state (equivalent to | ψ − 〉 = 1 2 ( | 01 〉 − | 10 〉 ) ), known as Bell’s

inequality, which must be fulfilled for any local realistic (hidden-variable) theory:

| P ( a → , b → ) − P ( a → , c → ) | − P ( b → , c → ) ≤ 1, (5)

where P ( a → , b → ) is the mean value of the product of the outcomes of measuring the spin components of two entangled spin 1/2 particles in the state | ψ − 〉 in the directions a → and b → respectively (being the possible outcomes ± 1 ), and analogously for P ( a → , c → ) and P ( b → , c → ) . The quantum-mechanical expectation value for P ( a → , b → ) is:

P ( a → , b → ) Q M = 〈 ψ − | σ → ⋅ a → ⊗ σ → ⋅ b → | ψ − 〉 = − cos θ   a → ,   b → , (6)

where θ   a → ,   b → is the angle between a → and b → . This means that the angles that produce maximal violation of inequality (5) are θ   a → ,   b → = θ   b → ,   c → = π / 3 ; that is, angles of 60˚ between a → , b → and b → , c → , and 120˚ between a → , c → (see Figure 10). According to Quantum Mechanics, the inequality should then be violated as 1.5 ≰ 1 .

Bell’s inequality (5) can be tested on the IBM 5Q computers using the circuits shown in Figure 11. The Hadamard gate and cNOT in the circuits from Figure 11 generate the state | ψ − 〉 from | 11 〉 . Then, the first circuit performs measurements in the X basis ( a → ) and in a basis whose vectors form a π / 3 angle with those of the X basis ( b → ), so it can be used to measure P ( a → , b → ) . The second and third circuits measure P ( a → , c → ) and P ( b → , c → ) , respectively. The results obtained using the ibmqx4 are shown in Figure 12.

The experimental values found for P ( a → , b → ) , P ( a → , c → ) and P ( b → , c → ) are [27] :

P ( a → , b → ) e x p = − 0.392 ± 0.014, P ( a → , c → ) e x p = 0.401 ± 0.009, P ( b → , c → ) e x p = − 0.389 ± 0.012, (7)

where P ( a → , b → ) is the sum of the probabilities of finding the results 00 or 11

minus the probabilities of finding 01 or 10, and similarly for P ( a → , c → ) and P ( b → , c → ) (note that the results according to Quantum Mechanics should be ±0.5). Hence, Bell’s inequality is violated:

1.182 ± 0.020 ≰ 1. (8)

Other Bell-type inequalities can be tested on the IBM 5Q as well; for instance, the CHSH inequality [28] [29] .

Mermin’s inequalities are a generalization of Bell-type inequalities for systems of more than two spin 1/2 particles (or qubits), derived in 1990 by Mermin [30] . The basic idea is that there exists an operator M_n (called a Mermin polynomial) whose quantum-mechanical expectation value exceeds the limits imposed by Local Realism for certain states. In this way, the incompatible predictions of the two theories can be experimentally tested and confronted: the inequalities must be fulfilled if Local Realism holds and violated in case Quantum Mechanics is correct.

The quantum-mechanical expectation value for M_n for some highly entangled states (GHZ-type states) exceeds the bounds imposed by Local Realism by an amount that grows exponentially with n (where n is the number of qubits). This fact implies that there is (in principle) no apparent limit to the amount by which Bell-type inequalities can be violated by certain entangled states. However, because the joint efficiency of n measurement apparatuses necessarily declines exponentially in n, and the n-qubit GHZ states are increasingly difficult to prepare as n grows, an exponentially greater violation of the inequalities for higher values of n will hardly be observed [30] . Mermin’s inequalities have been proposed as a figure of merit to assess the fidelity of a quantum computer [5] ; in fact, they can be tested on the IBM 5Q computers for n = 3 , 4 , 5 .

For n = 3 , a GHZ-type state and associated Mermin polynomial that give maximal violation of the corresponding inequality are:

| ψ 〉 = 1 2 ( | 000 〉 + i | 111 〉 ) , M 3 = σ y 2 σ x 1 σ x 0 + σ x 2 σ y 1 σ x 0 + σ x 2 σ x 1 σ y 0 − σ y 2 σ y 1 σ y 0 , 〈 M 3 〉 ≤ 2. (9)

For n = 4, those are [5] :

| ψ 〉 = 1 2   ( | 0000 〉 + e i π / 4 | 1111 〉 ) , M 4 = ( σ y 3 σ x 2 σ x 1 σ x 0 + σ x 3 σ y 2 σ x 1 σ x 0 + σ x 3 σ x 2 σ y 1 σ x 0 + σ x 3 σ x 2 σ x 1 σ y 0 )                     + ( σ y 3 σ y 2 σ x 1 σ x 0 + σ y 3 σ x 2 σ y 1 σ x 0 + σ y 3 σ x 2 σ x 1 σ y 0 + σ x 3 σ y 2 σ y 1 σ x 0                     + σ x 3 σ y 2 σ x 1 σ y 0 + σ x 3 σ x 2 σ y 1 σ y 0 ) − σ y 3 σ y 2 σ y 1 σ y 0 − σ x 3 σ x 2 σ x 1 σ x 0                     − ( σ y 3 σ y 2 σ y 1 σ x 0 + σ y 3 σ y 2 σ x 1 σ y 0 + σ x 3 σ y 2 σ y 1 σ y 0 + σ x 3 σ y 2 σ y 1 σ y 0 ) , 〈 M 4 〉 ≤ 4. (10)

And for n = 5:

| ψ 〉 = 1 2 ( | 00000 〉 + i | 11111 〉 ) , M 5 = ( σ y 4 σ x 3 σ x 2 σ x 1 σ x 0 + σ x 4 σ y 3 σ x 2 σ x 1 σ x 0 + σ x 4 σ x 3 σ y 2 σ x 1 σ x 0 + σ x 4 σ x 3 σ x 2 σ y 1 σ x 0                     + σ x 4 σ x 3 σ x 2 σ x 1 σ y 0 ) − ( σ y 4 σ y 3 σ y 2 σ x 1 σ x 0 + σ y 4 σ y 3 σ x 2 σ y 1 σ x 0 + σ y 4 σ y 3 σ x 2 σ x 1 σ y 0                     + σ y 4 σ x 3 σ y 2 σ y 1 σ x 0 + σ y 4 σ x 3 σ y 2 σ x 1 σ y 0 + σ y 4 σ x 3 σ x 2 σ y 1 σ y 0 + σ x 4 σ y 3 σ y 2 σ y 1 σ x 0                     + σ x 4 σ y 3 σ y 2 σ x 1 σ y 0 + σ x 4 σ y 3 σ x 2 σ y 1 σ y 0 + σ x 4 σ x 3 σ y 2 σ y 1 σ y 0 ) + σ y 4 σ y 3 σ y 2 σ y 1 σ y 0 , 〈 M 5 〉 ≤ 4. (11)

The circuits required for testing Mermin’s inequality for n = 3 and the choice of settings (9) are shown in Figure 13 (circuits for n = 4 and n = 5 are analogous). The Hadamard gate, cNOTs and S gate in Figure 13 generate the state

| ψ 〉 = 1 2 ( | 000 〉 + i | 111 〉 ) .

Changing the position of the S † gates in Circuit 1 from Figure 13 allows to measure σ x 2 σ y 1 σ x 0 and σ x 2 σ x 1 σ y 0 . However, from our own results and those obtained in [5] , it seems safe to assume symmetry under qubit exchange. That is:

〈 σ y 2 σ x 1 σ x 0 〉 e x p ≃ 〈 σ x 2 σ y 1 σ x 0 〉 e x p ≃ 〈 σ x 2 σ x 1 σ y 0 〉 e x p , and only one circuit is needed to

measure the three terms in the polynomial. The same is true for n = 4 and n = 5, which makes it possible to use a single circuit to measure all the terms with a same number of σ_ys.

The experimental results obtained for the states and polynomials (9), (10) and (11) on the ibmqx 2 are collected in Table 3. These results show a clear violation of Mermin’s inequalities for n = 3, 4, 5. They can be compared to those found by Alsina and Latorre [5] , also shown in Table 3 (the choice of state and polynomial for n = 5 is slightly different in [5] , but completely equivalent to (11),

since it has the same quantum-mechanical expectation value and Local Realism bound). We interpret the better results presented here to reflect the improvements of the IBM chips during the months in between both publications [31] .

The prime state | p n 〉 is the superposition of all the computational-basis states that correspond to prime numbers (written in binary format) up to a certain value N = 2 n − 1 [32] :

| p n 〉 = 1 π ( 2 n ) ∑ x ∈ prime 2 n − 1 | x 〉 , (12)

where π ( x ) , known as the prime counting function, is the number of primes less or equal than x. This state bears a large amount of entanglement [33] . The prime state for n = 3, i.e.

| p 3 〉 = 1 2 ( | 2 〉 + | 3 〉 + | 5 〉 + | 7 〉 ) = 1 2 ( | 010 〉 + | 011 〉 + | 101 〉 + | 111 〉 ) ,

can be created with the circuit shown in Figure 14.

The X gate acting on the first qubit ensures that all number states in the superposition created with the two Hadamard gates are odd: after these three gates have acted, the system is in the state

| ψ 〉 = 1 2 ( | 001 〉 + | 011 〉 + | 101 〉 + | 111 〉 ) .

Thus, all that remains is converting the term | 001 〉 (1 is not a prime number) into | 010 〉 . This is achieved with the help of a Toffoli gate and a cNOT.

After implementing the circuit of Figure 14 for creating the prime state | p 3 〉 on the ibmqx4, a joint measurement of the three qubits in the computational basis was carried out in order to assess the overall performance of the circuit. The results are shown in Figure 15.

The results show that, upon measurement of the purported prime state | p 3 〉 , a prime number is obtained with probability 72.5 ± 0.7 % [34] . In this sense it can be said that the prime state | p 3 〉 has been constructed with an approximated accuracy of 73%.

The interest in constructing prime states (something that can in principle be done efficiently using Grover’s algorithm with a primality test as oracle) is that

they would allow to experimentally test, for instance, Riemann hypothesis?one of the mathematical problems of the millennium [33] . But in order to falsify Riemann hypothesis (or extend the limits of its validity) it is necessary to build superposition states of billions of prime numbers. Constructing the state | p 3 〉 , even if this is not done by a general method for creating prime states, thus seems a modest first step.

Moreover, there exist a number of quantities in Number Theory that can surprisingly be measured experimentally on prime states [32] . The mean value of σ z 1 is:

〈 σ z 1 〉 = π 4 , 1 ( N ) − π 4 , 3 ( N ) − 1 π ( N ) , (13)

where π 4 , 1 ( N ) is the number of primes less or equal than N that can be written as 4 m + 1 with m a positive integer (i.e. that are equal to 1 (mod 4)) and π 4 , 1 ( N ) is the number of primes less or equal than N that can be written as 4m + 3 with m a non-negative integer (i.e. that are equal to 3 (mod 4)). Thus π ( N ) = π 4 , 1 ( N ) + π 4 , 3 ( N ) + 1 . The difference between these two quantities Δ ( N ) ≡ π 4 , 3 ( N ) − π 4 , 1 ( N ) is known as the Chebyshev bias, and it is curiously positive for most values of N for which it has been calculated so far [35] (the name is due to Pafnuty Chebyshev, who first noticed that the remainder upon dividing the primes by 4 gives 3 more often than 1).

Therefore, measuring the second qubit in the computational basis, and taking an outcome 0 as a 1 and an outcome 1 as a −1 when computing the mean value, give the Chebyshev bias, provided that π ( N ) is known. For the state | p 3 〉 created on the ibmqx 4, the experimental result for 〈 σ z 1 〉 obtained is:

〈 σ z 1 〉 e x p = − 0.301 ± 0.007, (14)

while the theoretical expected value for 〈 σ z 1 〉 is:

〈 σ z 1 〉 t h = − 0.500, (15)

which gives a relative error of around 40% in the measurement. Other quantities that can be measured experimentally on the prime state are:

〈 σ x 1 〉 = 2 π 2 ( 1 ) ( N ) π ( N ) ,         〈 σ x 1 σ x 2 + σ y 1 σ y 2 〉 = 4   π 2 ( 3 ) ( N ) π ( N ) , (16)

where π 2 ( 1 ) ( N ) is the number of twin prime pairs ( p , p + 2 ) less or equal than N with p = 1 (mod 4) and π 2 ( 3 ) ( N ) is the number of twin prime pairs ( p , p + 2 ) less or equal than N with p = 3 (mod 4). The sum π 2 ( 1 ) ( N ) + π 2 ( 3 ) ( N ) is equal to the number of twin prime pairs π 2 ( N ) . In analogy with the Chebyshev bias the twin prime bias is defined as Δ 2 ( N ) = π 2 ( 3 ) ( N ) − π 2 ( 1 ) ( N ) .

The experimental results of the measurements of operators (16) on | p 3 〉 on the ibmqx 4 are:

〈 σ x 1 〉 e x p = 0.435 ± 0.006, 〈 σ x 1 σ x 2 + σ y 1 σ y 2 〉 e x p = 0.641 ± 0.022, (17)

while the theoretical expected values are:

〈 σ x 1 〉 t h = 0.500,         〈 σ x 1 σ x 2 + σ y 1 σ y 2 〉 t h = 1.000   , (18)

which give relative errors of approximately 13% and 36% respectively.