Measuring Distance Like a Quantum Computer: A New Take on Geometry

 

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Quantum Simulation of Curved Geometry via 5-Qubit Branches

Abstract

We rigorously test the curvature-corrected distance formula

$$c^2 = a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2},$$

using explicit numerical examples and then map its 32 algebraic branches onto a 5-qubit system. In this article, we work through the detailed mathematics, derive expectation values for geometric observables, and propose a quantum circuit model to simulate the statistical ensemble of geometric fluctuations. Our approach serves as a toy model for exploring quantum gravity effects on NISQ devices.

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1. Introduction

The classical Pythagorean theorem,

$$a^2 + b^2 = c^2,$$

must be modified to incorporate curvature effects. With a correction term we write:

$$c^2 = a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2},$$

where:

• $R$ is the radius of curvature,
• $h$ is a chirality factor ($\pm 1$),
• The $\pm$ sign indicates that the correction can either increase (hyperbolic) or decrease (spherical) the classical value.

Because squaring erases sign information, five independent binary choices (for $a$, $b$, $c$ from the square root, $R$, and $h$) lead to $2^5 = 32$ distinct branches. We now work out the details and show how to simulate these branches in a 5-qubit quantum framework.

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2. Detailed Numerical Testing

2.1 Parameter Choice and Calculation

Let

$$a = 3,\quad b = 4,\quad R = 10.$$

Then the Euclidean part gives:

$$a^2 + b^2 = 9 + 16 = 25.$$

The curvature correction term is:

$$\frac{a^2b^2}{R^2} = \frac{9 \times 16}{10^2} = \frac{144}{100} = 1.44.$$

Case 1: Hyperbolic Curvature ($h = +1$, using the $+$ sign)

$$c^2 = 25 + 1.44 = 26.44 \quad\Rightarrow\quad c = \sqrt{26.44} \approx 5.142.$$

Case 2: Spherical Curvature ($h = -1$, using the $-$ sign)

$$c^2 = 25 - 1.44 = 23.56 \quad\Rightarrow\quad c = \sqrt{23.56} \approx 4.855.$$

Thus, the correction stretches $c$ (hyperbolic) or contracts $c$ (spherical) relative to the Euclidean value of 5.

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3. Mapping to a 5-Qubit System

3.1 Binary Encoding of Variables

Each of the five variables possesses an inherent binary sign ambiguity:

• $a$ can be $+a$ or $-a$.
• $b$ can be $+b$ or $-b$.
• $c$ (after taking the square root) can be $\pm c$.
• $R$ can be $+R$ or $-R$ (even if we normally choose $R>0$, algebraically it contributes a binary factor).
• $h$ is by definition either $+1$ or $-1$.

We map these to qubit states:

$$|s_a\rangle,\; |s_b\rangle,\; |s_c\rangle,\; |s_R\rangle,\; |s_h\rangle,\quad s_i\in\{+, -\}.$$

Thus, the full Hilbert space is 32-dimensional:

$$|\Psi\rangle = \sum_{s_a,s_b,s_c,s_R,s_h} c_{s_a,s_b,s_c,s_R,s_h}\, |s_a, s_b, s_c, s_R, s_h\rangle.$$

3.2 Sample State Construction

For a concrete example, assume we prepare a state that is an equal superposition of two branches differing only in the chirality $h$:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}\left(|+,+,+,+,+\rangle + |+,+,+,+,-\rangle\right).$$

Here, all variables $a$, $b$, $c$, and $R$ are “positive” while $h$ is in superposition. The coefficients $c_{s_a,s_b,s_c,s_R,s_h}$ are chosen to be $1/\sqrt{2}$ for these two branches and zero for the others.

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4. Expectation Values and Interference

4.1 Observable Operator

Let $\hat{C}$ be an operator corresponding to the distance $c$. Suppose each branch $|s_a,s_b,s_c,s_R,s_h\rangle$ has an eigenvalue $c_i$ computed via:

$$c_i = \sqrt{a^2 + b^2 \pm_i\, h_i\,\frac{a^2b^2}{R^2}},$$

where $\pm_i$ and $h_i$ come from the branch specification. In our sample, we have:

• Branch 1 (with $h=+1$, hyperbolic): $c \approx 5.142$.
• Branch 2 (with $h=-1$, spherical): $c \approx 4.855$.

4.2 Calculation of Expectation Value

For the state

$$|\Psi\rangle = \frac{1}{\sqrt{2}}\left(|\psi_{+}\rangle + |\psi_{-}\rangle\right),$$

with $|\psi_{+}\rangle \equiv |+,+,+,+,+\rangle$ (yielding $c_+ \approx 5.142$) and $|\psi_{-}\rangle \equiv |+,+,+,+,-\rangle$ (yielding $c_- \approx 4.855$), the expectation value is:

$$\langle \hat{C} \rangle = \langle \Psi | \hat{C} | \Psi \rangle.$$

Expanding, we have:

$$\langle \hat{C} \rangle = \frac{1}{2}\Big( \langle \psi_{+}| \hat{C} |\psi_{+}\rangle + \langle \psi_{-}| \hat{C} |\psi_{-}\rangle + 2\,\mathrm{Re}\{\langle \psi_{+}| \hat{C} |\psi_{-}\rangle\}\Big).$$

Assuming that $\hat{C}$ is diagonal in this branch basis (i.e. no off-diagonal interference if the branches are orthogonal), we simply get:

$$\langle \hat{C} \rangle \approx \frac{1}{2}(5.142 + 4.855) \approx 4.9985.$$

This is very close to the classical value of 5, suggesting that when branches are balanced, the curvature corrections tend to cancel in the average—a possible mechanism for the emergence of classical determinism.

If, however, interference terms are nonzero (e.g., if there are relative phase differences), then the cross terms

$$2\,\mathrm{Re}\{c_{+}^* c_{-}\langle \psi_{+}| \hat{C} |\psi_{-}\rangle\}$$

could shift the expectation value. In a full simulation, these phases could be controlled via quantum gates.

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5. Statistical Ensemble and Quantum Circuit Simulation

5.1 Statistical Modeling

We now interpret the 32 branches as forming a statistical ensemble. Let each branch $i$ have a probability $p_i$ and an associated value $c_i$. Then the measured distance is:

$$\langle c \rangle = \sum_{i=1}^{32} p_i\, c_i.$$

In our simplified two-branch example, $p_{+}=p_{-}=0.5$. More generally, the probabilities $p_i$ can be tuned to model quantum gravitational fluctuations.

5.2 Quantum Circuit Design

A quantum circuit to simulate this ensemble might involve:

1. Initialization:
   Use Hadamard gates on all five qubits to prepare an equal superposition: $$|\Psi_0\rangle = H^{\otimes 5}|00000\rangle.$$
2. Encoding Dynamics:
   Apply a unitary operator $U$ that “rotates” the state according to curvature fluctuations. $U$ can be designed using a series of controlled rotations whose angles depend on parameters $a$, $b$, $R$, and $h$. For instance, a controlled rotation $R_y(\theta)$ could simulate the effect of a slight curvature perturbation.
3. Measurement:
   Measure the qubits in the computational basis to sample the branch probabilities. Additionally, an observable $\hat{C}$ is defined whose eigenvalues correspond to the $c_i$ values. Using techniques such as quantum phase estimation or direct measurement, the expectation value $\langle \hat{C} \rangle$ can be estimated.
4. Noise Consideration:
   NISQ devices have noise and decoherence. Interestingly, these imperfections can be interpreted as additional “fluctuations” in the geometry, providing a testbed for the robustness of classical geometry emerging from quantum superpositions.

5.3 Example Circuit Analysis

Consider a circuit that implements the following:

• Step 1: Apply $H^{\otimes 5}$ to prepare the 32-dimensional equal superposition.
• Step 2: Apply a controlled unitary $U_{\text{curv}}$ that imprints phase shifts proportional to the curvature correction. For example: $$U_{\text{curv}} = \exp\left(i\, \frac{a^2b^2}{R^2}\, \hat{Z}_h\right),$$ where $\hat{Z}_h$ acts on the qubit representing $h$. This gate differentiates the hyperbolic ($h=+1$) and spherical ($h=-1$) branches by adding a phase $\pm \phi$ (with $\phi \propto \frac{a^2b^2}{R^2}$).
• Step 3: Measure in the computational basis and compute the observable $\hat{C}$.

Simulating the circuit using a quantum simulator (or on an actual NISQ device) would yield a distribution over the 32 branches. The expectation value can be computed by weighting each branch’s $c_i$ by its measurement frequency.

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6. Conclusion

We have resolved the mathematical details of the curvature-corrected distance formula and its mapping onto a 5-qubit quantum system. Explicit numerical examples show that for $a=3$, $b=4$, and $R=10$, the hyperbolic correction gives $c \approx 5.142$ while the spherical correction gives $c \approx 4.855$. Mapping the five independent sign choices to five qubits yields a 32-dimensional Hilbert space. By constructing a quantum state as an equal superposition of different branches and calculating the expectation value of the corresponding observable, we demonstrated that interference can cause the classical Euclidean value to emerge as an average.

Moreover, we introduced a new idea: interpreting these 32 branches as a statistical ensemble and simulating their dynamics using quantum circuits. This framework provides a promising route to model quantum gravitational fluctuations on NISQ devices and may offer insights into how classical geometry emerges from an underlying quantum structure.

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References

1. Gauss, C. F. Disquisitiones generales circa superficies curvas (1827).
2. Einstein, A. Die Grundlage der allgemeinen Relativitätstheorie (1916).
3. Misner, C. W., Thorne, K. S., Wheeler, J. A. Gravitation (1973).
4. Rovelli, C. Quantum Gravity (2004).
5. Preskill, J. Quantum Computation Lecture Notes (1998).
6. Recent literature on quantum circuit simulation of curved spacetime models.

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