Loading Probability Distributions in a Quantum circuit

One can find some papers written on generating probability distributions. One of the earliest papers on generating probability distributions is by [5]. In this paper, the author gives a scheme on how to generate a superposition of quantum states of the form given in (1), that resembles a given probability distribution.

The method followed in this scheme is an incremental expansion scheme, whereby, at every point in the expansion, the system is expanded by one qubit by a controlled rotation. The existing qubits form the control set and the target qubit is the newly added qubit. This can be better understood by considering that every state of the existing system will expand by the addition (tensor product) of a $\lvert{0}\rangle$ and a $\lvert{1}\rangle$ to its left as shown in equation (2).

Fig. 1 illustrates the expansion scheme by taking the example of a transition from a 1 qubit system to a 2 qubit system. This method requires $2^{n-1}$ controlled $R_{Y}$ gates and $\approx 2^{n}$ $X$ gates, for a $n$ qubit system.

In Fig. 1, the variables $P_{0},P_{1},P_{00}$, etc., denote the probability amplitudes and not the probabilities. A more detailed explanation can be found in [6].

In [11], the authors use a Quantum Generative Adversarial Networks (QGANs) to learn distributions from given training data. The algorithm uses a hybrid quantum-classical computation approach, where a quantum computer generates a probability distribution and a classical computer tries to discriminate between the generated distribution and the distribution of the training data. The quantum computer generates a probability distribution using variational circuits. The variational circuit uses $R_{Y}$ rotation gates and controlled $Z$ ($CZ$) gates for entanglement. Fig. 2 shows one layer of the variational circuit in a 4 qubit system.

For a $n$ qubit system, this method requires an initial set of $n$ rotation parameters and then for every layer we have $n$ set of parameters for the $n$ number of $R_{Y}$ gates, and $n-1$ or $n$ number of $CZ$ gates depending upon the type of entanglement (linear or circular). Overall, it requires $nl+n$ parameters for $l$ layers of the parameterized circuit. To begin with, the variational method uses a normal distribution against which the generated distribution is compared. The loss function used is $\|.\|_{2}$ of the difference between the generated and the desired distributions. For the discriminator the training data is sampled from log-normal distributions.

In [9], the authors give a quantum circuit scheme to prepare arbitrary quantum states. The authors were able to reduce the upper bound on the number of CNOT gates required for an even number of qubits. The authors also show that some part of circuit computation can be performed in parallel, thus reducing the computational depth. In [4], the authors discuss the evolution of quantum walks as a function of time over a graph. Under certain conditions, the limiting distribution of the evolution is uniform.

In this article, we propose a method to generate symmetrical and asymmetrical probability distributions by studying the trajectory of the individual quantum states under the action of rotation and entanglement gates. The objective is to generate distributions that does not require circuit elements that grow exponentially with the number of qubits. The approach exploits symmetry properties of distribution curves to do away with redundant rotation and controlled gates. Variational solvers are used to fix the rotation parameters of the gates, with some constraints, such that a finer 1:1 correspondence between the probability of occurrence of a state and points in the desired probability distribution curve is obtained. The ideas given here could be used as a general framework to generate a variety of probability distribution curves.

The rest of the paper is organized as follows. In section 2, we discuss the basic concept that the methodology proposed here uses. It discusses the trajectory of a quantum state as qubits are added and unitary operations in the form of gates get applied. This section also discusses ways to generate symmetric and asymmetric (skewed) distributions. In section 3, we discuss the algorithm to generate a distribution using a variational solver. In section 4 we present some results that were obtained using both simulators and quantum hardwares. Section 5 gives the conclusions.

In Fig. 3, we can see that if $\theta_{1}=\frac{\pi}{2}$ radians ($n=0$ in (8)), the probability amplitudes corresponding to $\lvert{00}\rangle$ and $\lvert{10}\rangle$ will become equal with a value $\frac{1}{\sqrt{2}}\cos{(\theta_{2}/2)}$, while those corresponding to $\lvert{01}\rangle$ and $\lvert{11}\rangle$ will also become equal with a value $\frac{1}{\sqrt{2}}\sin{(\theta_{2}/2)}$. To have a probability distribution symmetric around the centre, we could exchange the probability amplitudes of $\lvert{10}\rangle$ and $\lvert{11}\rangle$, as shown in Fig. 5.

The exchange of the probability amplitudes of the states $\lvert{10}\rangle$ and $\lvert{11}\rangle$ can be achieved by the application of an $X$ Pauli gate, conditioned on the first qubit (MSB) being in state $\lvert{1}\rangle$. This is the CNOT gate as shown in Fig. 5. The bar plot in the figure is for a $\theta_{2}=\frac{2\pi}{3}$ radians. A CNOT conditioned on $\lvert{0}\rangle$ in the first qubit would have resulted in a symmetric distribution where the corner amplitudes are higher than the amplitudes at the centre (exchange of probability amplitudes of the states $\lvert{00}\rangle$ and $\lvert{01}\rangle$). To generate distributions that have central tendencies, we will first have to figure a suitable value for the rotation angles of the lower significant bits, and then have to decide the control state of the MSB over which the states are flipped. To generate the kind of distribution seen in Fig. 5, we must have $\cos{(\theta_{2}/2)}<\sin{(\theta_{2}/2)}$. In other words, we must have $\frac{5\pi}{2}>\theta_{2}>\frac{\pi}{2}$.

Let us now add a third qubit to the system. The numbering scheme of the qubits will then change. The qubit corresponding to the MSB will become $q_{2}$. The lower significant bits will then become $q_{1}$ and $q_{0}$. The third qubit is then passed through a rotation gate $R_{Y}$ with parameter $\theta_{3}$. Addition of the third qubit will result in the expansion of the number of states to eight. The $R_{Y}(\theta_{3})$ operation on the third qubit will result in each of the four existing states of the 2 qubit system dividing up with an extra $\cos{(\theta_{3}/2)}$ and $\sin{(\theta_{3}/2)}$ terms as shown in Fig. 6.

In Fig. 6, the $\cos{(\theta_{1}/2)}$ and $\sin{(\theta_{1}/2)}$ terms have been taken out as $\frac{1}{\sqrt{(}2)}$, since $\theta_{1}=\frac{\pi}{2}$. If we look at the states after the application of $R_{Y}(\theta_{3})$ to the third qubit, we can see that if the amplitudes of the states $\lvert{100}\rangle$ and $\lvert{101}\rangle$ followed by $\lvert{110}\rangle$ and $\lvert{111}\rangle$ are exchanged, the distribution of the amplitudes are symmetrical. This exchange can happen if conditioned on the qubit corresponding to the MSB, i.e., $q_{2}$, being in state $\lvert{1}\rangle$, a CNOT gate is applied to the qubit corresponding the lowest significant bit (LSB), i.e., $q_{0}$. This is depicted in Fig. 7.

The bar plot in Fig. 7 was created with $\theta_{2}=\frac{2\pi}{3}(120^{\circ})$ and $\theta_{3}=100^{\circ}$. From the figure it is clear that by exchanging the amplitudes of states $\lvert{100}\rangle$ and $\lvert{101}\rangle$ and further in $\lvert{110}\rangle$ and $\lvert{111}\rangle$, a symmetric distribution can be achieved.

The distribution as seen in Fig. 7 resembles a Gaussian distribution. This is a distribution that has monotonically decreasing amplitudes on either side of the centre. Not all symmetric distributions need to have this feature. However, if we are to model such distributions, it is important that we find the criteria that the parameters ($\theta_{2}$ and $\theta_{3}$ in this case) must satisfy. Let us look at the amplitudes where MSB is $0$. If we look at the states $\lvert{000}\rangle$ and $\lvert{001}\rangle$, the amplitude of $\lvert{001}\rangle$ will be more than $\lvert{000}\rangle$, only if $\cos{(\theta_{3}/2)}\leq\sin{(\theta_{3}/2)}$. This condition will also ensure that the amplitude of $\lvert{011}\rangle$ is more than the amplitude of $\lvert{010}\rangle$. Another condition that needs to be satisfied is that the amplitude of $\lvert{010}\rangle$ is more than the amplitude of $\lvert{001}\rangle$. That will happen if the following condition is satisfied.

To put it all together, the conditions that the parameters need to obey such that we have a distribution which has a central tendency with a monotonically decreasing function on either side, are given as follows.

For an $n$ qubit system, we will need to satisfy the constraints given in (9).

To summarize, to have a symmetric distribution we need to have $\theta_{1}=\frac{\pi}{2}$ and circuits of the type given in Fig. 7. To generate distributions that have a central tendency, like in Gaussian distribution curves, constraints of the type given in (9) need to be met. If we are to match the generated distribution curve to a given distribution curve, we could use variational solvers to do that. This is discussed in section 3. For an $n$ qubit system, the above circuit construction requires $n$ $R_{Y}$ rotation gates and $n-1$ CNOT gates. When used with a variational solver, we will need $n-1$ parameters.

When we say asymmetric distributions, we mean distributions that have skewness in the distributions. It could be distributions like skew-normal, log-normal distributions etc. We approach this problem by dividing the distribution into two areas, based on the MSB. Positive skewness indicates that the mode of the distribution lies on the side where the qubit corresponding to the MSB has the state $\lvert{0}\rangle$. The opposite ($\lvert{1}\rangle$) is true for negative skewness. Distributions with positive skewness will have the monotonically decreasing tail on the side where the MSB qubit has state $\lvert{1}\rangle$. This part (MSB with state $\lvert{1}\rangle$) can be handled by following the approach given in (9). However, the inequalities in (9) will get reversed as given in (10). Conditioned on the MSB qubit in state $\lvert{1}\rangle$, the parameter values of the $R_{Y}$ rotation gates on the lower significant bits are progressively increased.

It is to be noted here that, $\theta_{1}$ will depend upon the nature of the skewness. A positive skewness would require $\theta_{1}\leq\frac{\pi}{2}$ ($\cos{\theta_{1}}\geq\sin{\theta_{1}}$). For negative skewness the inequalities need to be reversed.

Fig. 8(a) gives the circuit of such an implementation in a 5 qubit system. Fig. 8(b) gives the count statistics of the states that have MSB qubit as $\lvert{1}\rangle$, after the execution of the circuit for 10000 runs in a Qasm simulator [2]. The angles considered for the circuit were $\theta_{1}=\pi/3~{}(60^{\circ}),~{}\theta_{2}=0.361\pi~{}(65^{\circ}),~{}\theta_{3}=0.416\pi~{}(75^{\circ}),~{}\theta_{4}=0.444\pi~{}(80^{\circ}),~{}\theta_{5}=0.472\pi~{}(85^{\circ})$.

The other side of the distribution pertaining to the MSB qubit at state $\lvert{0}\rangle$, will have the state with the peak probability amplitude. It will also have adjacent states with falling amplitudes. Fig.9 shows the part of a log-normal distribution ($\mu=0,\sigma=0.5$) where the MSB qubit has state $\lvert{0}\rangle$. The distribution is for a 5 qubit system having a total of 32 data points (16 with MSB as $\lvert{0}\rangle$ and 16 with MSB as $\lvert{1}\rangle$).

From the figure, it can be seen that there is some amount of symmetry here. In the symmetric distribution, we had seen the symmetry is around the median and peak amplitude. This is also the place where the the MSB qubit state changes from $\lvert{0}\rangle$ to $\lvert{1}\rangle$. In the asymmetric or the skew-symmetric case, however, the peak probability amplitude (around which there is some symmetry) can be found around the point where the second most significant qubit changes from state $\lvert{0}\rangle$ to $\lvert{1}\rangle$. We could then follow the method we used while generating symmetric distributions. This however has to be conditioned on the MSB qubit in state $\lvert{0}\rangle$. Fig. 10(a) gives the circuit construction for a 5 qubit system. This is identical to the construction given in Fig. 7, except that all the $R_{Y}$ gates are now controlled and acts only when the MSB qubit is in state $\lvert{0}\rangle$. The two $X$ gates ensure that the control is on $\lvert{0}\rangle$.

Now that the symmetry is being considered only on one side, the control qubit for the CNOT gates should be the second MSB qubit, i.e., $q_{3}$ in Fig. 10(a). The constraints on the angles could be as given in (9), except that the indices $j$ and $k$ will now start from 3 instead of 2. Note that we are not explicitly making the assignment $\theta_{2}=\frac{\pi}{2}$, since the distribution is not an exact symmetry. The finer adjustments and estimation of the rotation angle parameters may be done by the variational solver.

The construction in Fig. 10(a) would mainly lead to a somewhat symmetric kind of a distribution. To bring in an extra layer of asymmetry we could add some more $R_{Y}$ gates but controlled on the two most significant qubits, i.e. $q_{4}$ and $q_{3}$ in a 5 qubit system, as shown in Fig. 10(b). In the figure, the two $X$ gates on $q_{3}$ ensure that the extra layer of controlled rotations happen only when the two most significant qubits are in the state $\lvert{00}\rangle$. This will fine tune the probability amplitudes for the states $\lvert{00000}\rangle,\lvert{00001}\rangle\ldots\lvert{00111}\rangle$. We could very well do these kind of fine tuning when the two most significant qubits are in the state $\lvert{01}\rangle$. This will be dependent on the type of distribution that we are trying to match or train our circuit for. A highly skewed distribution may require either/or of such fine adjustments. Our final circuit construction for the asymmetric distribution is given in Fig. 11.

For a $n$ qubit system, this circuit construction would need the following.

• •

  1 $R_{Y}$ rotation gate,

• •

  $2(n-1)$ controlled $R_{Y}$ rotation gates,

• •

  $n-2$ doubly controlled rotation gates,

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  $n-2$ CNOT gates,

• •

  $4$ Pauli $X$ gates.

A variational solver for this circuit would need $3n-3$ parameters.

While the technique proposed here may be able to generate distributions that are close to the actual desired ones, there are some drawbacks to it. One primary drawback to this method is the presence of CNOT and $R_{Y}$ gates controlled on some significant qubit. In a large circuit system having a number of qubits, implementing this kind of an architecture in a real hardware will require a lot of swap gates. The most significant qubits will have to be positioned in the hardware in such a way that it has the maximum number of connections with other qubits. Wherever, connections are lacking, swap gates come into the picture resulting in increased gate counts and circuit depth. In other words, we will need a highly interconnected hardware to scale the method proposed here. The circuit proposed in [11] does not suffer from this drawback as the controlled gates require adjacent connections.

The second major drawback of this method is the number of controlled gates used. Controlled gates like CNOT, controlled $R_{Y}$ gates are prone to noise. CNOT gates also come into the picture when swap gates are used. With a large number of such gates, the results obtained will have errors. This is especially true for the circuits generating asymmetric distributions. If we are to implement a larger system (more states representing probability distributions), in a real hardware, we will need gate noise error mitigation methods to be implemented as well.