Noise analysis for the Sorkin and Peres tests performed on a quantum computer

Abstract

We use quantum computers to test the foundations of quantum mechanics through quantum algorithms that implement some of the experimental tests as the basis of the theory’s postulates. These algorithms can be used as a test of the physical theory under the premise of perfect hardware or as a test of the hardware under the premise that quantum theory is correct. In this work, we show how the algorithms can be used to test the efficacy of a quantum computer in obeying the postulates of quantum mechanics. We study the effect of different types of noise on the results of experimental tests of the postulates. A salient feature of this noise analysis is that it is deeply rooted in the fundamentals of quantum mechanics as it highlights how systematic errors affect the quantumness of the quantum computer.

Appendices

Appendices

A Generating random states

The values of parameter $\theta$ and $\varphi$ for the preparation of random states are generated as follows. We take a random variable r uniformly distributed over the interval [0, 1] transforming it appropriately to get the required probability distributions for $\theta$ and $\varphi$. Let the required functions be $\theta (r)$ and $\varphi (r)$. Then,

$$\begin{aligned} P(\theta ) d\theta =&P(r) dr \nonumber \\ \implies ~ \int \limits _{0}^{\theta } P(\theta ) d\theta =&\int \limits _{0}^{r} P(r) dr \nonumber \\ \implies ~ \int \limits _{0}^{\theta } \frac{\sin \theta }{2} d\theta =&\int \limits _{0}^{r} 1 dr \nonumber \\ \implies ~ \frac{1 - \cos \theta }{2} =&r \nonumber \\ \implies ~ \theta (r) =&\cos ^{-1}\left( 1 - 2r\right) \end{aligned}$$

(16)

Now, we can generate a random number r uniformly over [0.1] and calculate $\theta (r)$ with the required distribution. Similarly,

$$\begin{aligned} P(\varphi ) d\varphi =&P(r) dr \nonumber \\ \implies ~ \int \limits _{0}^{\varphi } P(\varphi ) d\varphi =&\int \limits _{0}^{r} P(r) dr \nonumber \\ \implies ~ \int \limits _{0}^{\varphi } \frac{1}{2\pi } d\varphi =&\int \limits _{0}^{r} 1 dr \nonumber \\ \implies ~ \frac{\varphi }{2\pi } =&r \nonumber \\ \implies ~ \varphi (r) =&2 \pi r \end{aligned}$$

(17)

B Higher-dimensional Sorkin test

We prove by mathematical induction Sorkin’s parameter with 3 terms,

$$\begin{aligned} |x_1 + x_2 + x_3|^2 =&|x_1 + x_2|^2 + |x_1 + x_3|^2 + |x_2 + x_3|^2 \nonumber \\&-|x_1|^2 - |x_2|^2 - |x_3|^2 \end{aligned}$$

(18)

If the 3-term expression is true as above, then with 4 terms,

$$\begin{aligned} |x_1+x_2+x_3+x_4|^2 =&|x_1+x_2|^2 + |x_1+x_3|^2 + |x_1+x_4|^2\nonumber \\&+|x_2+x_3|^2 + |x_2+x_4|^2\nonumber \\&+|x_3+x_4|^2\nonumber \\&-2|x_1|^2 - 2|x_2|^2 - 2|x_3|^2 - 2|x_4|^2 \end{aligned}$$

(19)

If the 4-term expression is true as above, then with 5 terms,

$$\begin{aligned} |x_1+x_2+x_3+x_4+x_5|^2 =&|x_1+x_2|^2+|x_1+x_3|^2+|x_1+x_4|^2\nonumber \\&+|x_1+x_5|^2 +|x_2+x_3|^2+|x_2+x_4|^2 \nonumber \\&+|x_2+x_5|+|x_3+x_4|^2+|x_3+x_5|^2\nonumber \\&+ |x_4+x_5|^2-3|x_1|^2-3|x_2|^2-3|x_3|^2\nonumber \\&-3|x_4|^2-3|x_5|^2 \end{aligned}$$

(20)

Observing the trend, we can guess that for n terms,

$$\begin{aligned} \left| \sum \limits _{i=1}^{n}x_i\right| ^2 = \frac{1}{2}\sum \limits _{i,j}^{n,n}|x_i+x_j|^2 - (n-1)\sum \limits _{i}^{n}|x_i|^2 \end{aligned}$$

(21)

The general expression can be proven by mathematical induction.

For $n=1$,

LHS = $|x_1|^2$

RHS = $|x_1|^2$

Therefore, it holds for $n=1$. We assume that the expression is true for $n=k$. Now, for $n=k+1$,

$$\begin{aligned} \left| \sum \limits _{i=1}^{k} x_i + x_{k+1}\right| ^2 =&\left| \sum \limits _{i=1}^{k}x_i\right| ^2 + |x_{k+1}|^2 + \sum \limits _{i=1}^{k}2\Re \{x^*_i x_{k+1}\} \nonumber \\ =&\left| \sum \limits _{i=1}^{k}x_i\right| ^2 + |x_{k+1}|^2 + \sum \limits _{i=1}^{k}\left( |x_i + x_{k+1}|^2 \right. \nonumber \\&\left. - |x_i|^2 - |x_{k+1}|^2\right) \end{aligned}$$

(22)

Using Eq. (21):

$$\begin{aligned} \left| \sum \limits _{i=1}^{k} x_i + x_{k+1}\right| ^2 =&\frac{1}{2}\sum \limits _{i,j}^{k,k}|x_i+x_j|^2 - (k-1)\sum \limits _{i}^{k}|x_i|^2\nonumber \\&+ |x_{k+1}|^2 + \sum \limits _{i=1}^{k}\left( |x_i + x_{k+1}|^2 - |x_i|^2 - |x_{k+1}|^2\right) \nonumber \\ =&\frac{1}{2}\sum \limits _{i,j}^{k,k}|x_i+x_j|^2 \nonumber \\&+ \sum \limits _{i=1}^{k}|x_i + x_{k+1}|^2 + |x_{n+1}|^2 - k\sum \limits _{i}^{k}|x_i|^2 -k |x_{k+1}|^2\nonumber \\ \end{aligned}$$

(23)

Now, observe the fact that

$$\begin{aligned}&\frac{1}{2}\sum \limits _{i,j}^{k+1,k+1}|x_i+x_j|^2 \nonumber \\&\quad = \frac{1}{2}\sum \limits _{i,j}^{k,k+1}|x_i+x_j|^2 + \frac{1}{2}\sum \limits _{j}^{k+1}|x_{k+1}+x_j|^2 \nonumber \\&\quad = \frac{1}{2}\sum \limits _{i,j}^{k,k}|x_i+x_j|^2 + \frac{1}{2}\sum \limits _{i}^{k}|x_i+x_{k+1}|^2 + \frac{1}{2}\sum \limits _{j}^{k+1}|x_{k+1}+x_j|^2 \nonumber \\&\quad = \frac{1}{2}\sum \limits _{i,j}^{k,k}|x_i+x_j|^2 + \frac{1}{2}\sum \limits _{i}^{k}|x_i+x_{k+1}|^2 + \frac{1}{2}\sum \limits _{j}^{k}|x_{k+1}+x_j|^2 + |x_{k+1}|^2\nonumber \\&\quad = \frac{1}{2}\sum \limits _{i,j}^{k,k}|x_i+x_j|^2 + \sum \limits _{i=1}^{k}|x_i + x_{k+1}|^2 + |x_{k+1}|^2 \end{aligned}$$

(24)

Equation (23) simplifies to

$$\begin{aligned} \left| \sum \limits _{i=1}^{k} x_i + x_{k+1}\right| ^2 =&\frac{1}{2}\sum \limits _{i,j}^{k+1,k+1}|x_i+x_j|^2 - k\sum \limits _{i}^{k+1}|x_i|^2 \nonumber \\ \end{aligned}$$

(25)

Therefore, by mathematical induction, Eq. (21) is true for all $n\ge 1$.

With this generalized expression, we can define higher-dimensional Sorkin’s parameter

$$\begin{aligned} \kappa _n =&\left| \sum \limits _{i=1}^{n}x_i\right| ^2 - \frac{1}{2}\sum \limits _{i,j}^{n,n}|x_i+x_j|^2 + (n-1)\sum \limits _{i}^{n}|x_i|^2 \end{aligned}$$

(26)

which can be used to test Born’s rule in higher-dimensional systems. Rewriting it in a form suitable for programming,

$$\begin{aligned} \kappa _n =&\left| \sum \limits _{i=1}^{n}x_i\right| ^2 - \sum \limits _{i,j>i}^{n-1,n}|x_i+x_j|^2 + (n-2)\sum \limits _{i}^{n}|x_i|^2 \end{aligned}$$

(27)

C Statistical fluctuations

The quantities that are considered here depend on the probabilities of measurement outcomes. Therefore, even if the apparatus (in this case, the quantum computer) is ideal, there will be statistical fluctuations in the result due to limited number of trials in the experiment. For example, if we conduct N trials from which we estimate the value of $P_i = \left| \langle i|\psi \rangle \right| ^2$, the observed probability (the relative frequency of the outcome) will have a Gaussian distribution around the true probability value, with the mean of the distribution being the true probability value $P_i$ and the standard deviation being $\sqrt{P_i (1-P_i)/N_i}$. The fluctuation in the observed probability $\Delta P_i$ with about 95% confidence interval is given by

$$\begin{aligned} \Delta P_i =&\pm 1.96 \sqrt{\frac{P_i(1-P_i)}{N}} \end{aligned}$$

(28)

Fig. 6

Plot of error in the relative frequency of outcome as a function of the true probability. The number of trials is $10^4$

Figure 6 shows the error in the relative frequency of outcome as function of the true probability of that outcome, for $N=10^4$. And in terms of projective measurement outcomes,

$$\begin{aligned} \Delta P_i =&\pm 1.96 \sqrt{\frac{\left| \langle 0|\psi \rangle \right| ^2 \left| \langle 1|\psi \rangle \right| ^2}{N}} \end{aligned}$$

(29)

This is the maximum fluctuation of the measurement frequency about the true probability with 95% confidence interval. Another way is to take the percentile confidence interval. For example, if we want a 99-percentile confidence interval with our data, we can sort the data in ascending order and take the 0.005-percentile and 0.995-percentile as the lower and upper end of the confidence interval, respectively. This fluctuation in the observed probability will propagate to the quantities that we calculate using the probability values. To calculate the error in gamma values, we differentiate equation (4) to find the fluctuations in the value of $\gamma _{ij}$ with fluctuating measurement frequencies and consequently the fluctuations in the value of F.

D Plots with multiple states

Here, we present the plots with each kind of noise discussed in the main text with multiple randomly selected input states.

Fig. 7

a The variation of $\kappa$ in the presence of different amounts of readout error simulated for a set of random input states. (Different coloured dots correspond to different states.) A common feature of all the curves is that they are all quadratic in nature. Another notable feature is that $\kappa = 0$ for readout errors of $p=0$ and $p=0.5$ irrespective of the initial state. b The variation of F in the presence of different amounts of readout error simulated for a set of random input states. (Different coloured dots correspond to different states.) A common feature of all the curves is that there is a threshold of the amount readout error beyond which the superposition principle is violated. Below that threshold, the quantum computer behaves as if quantum mechanics is quaternionic

Fig. 8

a Variation of $\kappa$ with different amounts of depolarizing noise for a set of random states. (Different coloured dots represent different states.) b Variation of F with different amounts of depolarizing noise for a set of random states. (Different coloured dots represent different states.) Depolarizing noise does not cause a violation of the superposition. However, it does make the effective behaviour of the quantum computer quaternionic

1.1 D.1 Readout error

See Fig. 7.

1.2 D.2 Depolarizing error

See Fig. 8.

1.3 D.3 Thermal noise

See Fig. 9.

Fig. 9

a Variation of $\kappa$ with thermal relaxation noise for which $T_2 = 2T_1$. Born’s rule is violated for values of $T_1$ less than the gate times. But as $T_1$ becomes larger than all the gate times, the value of $\kappa$ approaches the ideal value of zero. b Variation of F with thermal relaxation noise for which $T_2 = 2T_1$. The plots for a set of random states (different colours correspond to different states) show that depending on the initial state, the superposition principle may be violated. But as $T_1$ becomes larger than all the gate times, the value of F approaches the ideal value of unity