Digital Quantum Simulation

• Digital quantum simulation (DQS) is a programmable approach that uses gate-based quantum computers to emulate the dynamics of complex quantum systems, including bosonic models.
• DQS leverages Trotter–Suzuki decompositions and Gray code mapping to efficiently encode infinite-dimensional bosonic states into finite qubit registers.
• Experimental implementations on IBM hardware validate DQS by reproducing interferometric complementarity and precise visibility measurements in bosonic interferometry.

Digital quantum simulation (DQS) is the use of universal, gate-based quantum computers to efficiently emulate the dynamics of complex quantum systems, including those that are classically intractable. By encoding the state of the target system into registers of qubits and reproducing its time evolution as a discrete sequence of quantum gates—typically via Trotter–Suzuki decompositions—DQS provides a flexible and programmable framework for simulating spin, fermionic, and bosonic models. Recent research has extended the scope of DQS to simulating fundamental quantum phenomena in cosmology, particle physics, quantum optics, and the foundations of quantum mechanics, including the direct digital simulation of bosonic systems and the quantum complementarity principle.

1. Mapping Bosonic Systems to Qubits via the Gray Code

Bosonic systems present a significant challenge for quantum simulation due to their infinite-dimensional local Hilbert spaces. To render such systems amenable to DQS, a truncation to a finite Fock subspace is performed and each occupation-number state is encoded as a bitstring. A recently developed mapping employs the Gray code, assigning each Fock state to a unique length-$N_q$ bitstring such that adjacent number states differ by only one bit; this property allows bosonic creation and annihilation operators to be represented efficiently as products of Pauli projectors and single-qubit raising/lowering operators. For example, given a Fock state $|n\rangle$ with binary representation $B_1B_2\ldots B_{N_q}$, the Gray code is constructed as $G_1=B_1$, $G_k=B_k\oplus B_{k-1}$ for $k\geq2$ (with $\oplus$ the bitwise XOR). Creation and annihilation operators act by flipping the appropriate bit and multiplying by products of Pauli $X$, $Y$, and $Z$ operators conditioned on the remaining bits, yielding a local and scalable mapping to quantum gates (Brasil et al., 3 Feb 2025).

This approach leads to the following operator structure:

• Local projector $\mathcal{P}_{G_k} = (1/2)(I \pm Z)$, with the $+$ ($-$) sign chosen for $G_k=0$ ($G_k=1$).
• Local "ladder" operator $\mathcal{Q}_{G_k} = (1/2)(X \mp i Y)$, again switching sign according to $G_k$. The overall operator for a mode is then a product of these elements, acting only on the bit that changes under a creation or annihilation.

2. Digital Quantum Circuits for Bosonic Interferometry and Complementarity

Using this Gray code mapping, researchers construct quantum circuits that simulate interferometric experiments involving bosons—such as the Afshar, Unruh, and Pessoa J interferometer variants. All the key linear optical elements—balanced and biased beam splitters (BS/BBS), phase shifters, and mirrors—are implemented as sequences of unitary gates on encoded qubits. For example, a balanced BS acting on modes $A$ and $B$ is mapped as $$U_{\text{BS}} = \exp(i\lambda(X \otimes X + Y \otimes Y))$$, with $\lambda$ related to the mixing angle; this form enables a decomposition into commuting exponentials.

Blockers (simulated absorptive or destructive measurements) are implemented by an in-circuit measurement and conditional reset, thus emulating the projective removal of bosons from certain modes. All measurements are repeated to build classical statistics analogous to photodetector counts (Brasil et al., 3 Feb 2025).

3. Simulation of Modern Interferometric Complementarity Experiments

DQS is applied to simulate the modified Unruh protocol (a two-level, Mach-Zehnder interferometer variant) and Pessoa J's extended Afshar interferometer, each probing complementarity in novel ways:

• In the Unruh experiment, phase shifters placed in either arm yield, via the circuit, probabilities at detectors that depend on the chosen phases. Varying one phase testifies to "corpuscular" (particle-like) or "wave-like" behavior based on the observed visibility $\mathcal{V} = (\max p - \min p)/(\max p + \min p)$.
• In Pessoa J's variant, biased beam splitters mimic partial "wire grid" absorption. Quantum circuits implement these by generalizing the BS unitary as $U_{\text{BBS}}$, and detection at extra output modes tracks which-path information and interference suppression in the presence of partial absorbers.

Results from IBM's Eagle r3 quantum processors show experimental outcome distributions and visibilities in precise agreement with theoretical predictions, confirming the correct behavior both in presence and absence of simulated blockers.

4. Quantitative Analysis of Quantum Complementarity

The simulated experiments are analyzed according to an updated quantum complementarity principle (QCP), which, rather than relying on retroactive or global interpretation, defines the tradeoff between interference and which-path information through quantitative complementarity relations: $\mathcal{W}(\rho) + \mathcal{P}(\rho) = 1$ where $\mathcal{W}$ is the l$_1$-norm coherence of the density matrix (measuring "waviness") and $\mathcal{P}$ its predictability ("particle-ness"). In practice, after a biased beam splitter with transmission $T_1$ and reflection $R_1$, the post-BS state for the quanton is $$| \psi_1 \rangle = T_1 |10\rangle + i R_1 |01\rangle$$, yielding $\mathcal{W} = 2T_1 R_1$ and $\mathcal{P} = 1-2T_1 R_1$ (Brasil et al., 3 Feb 2025).

Digital quantum simulations verify that these relations hold at each stage of the interferometric evolution. Notably, they demonstrate that no violation of Bohr’s complementarity arises when examining specific quantum state preparations or measurement results, even in configurations previously alleged to challenge complementarity.

5. Experimental Implementation and Simulated Results

Circuit depth and gate counts are determined by the number of modes, truncation level, and complexity of the interferometer. For simple two-mode interferometers, circuits are shallow and manageable with current quantum processors. The Gray code mapping is especially advantageous, minimizing gate count and reducing the required number of qubits.

On real hardware (such as IBM Sherbrooke or Kyiv), measured probabilities for detectors and blocked modes match theoretical quantum predictions, including quantified reductions corresponding to experimentally relevant photon losses (e.g., the $\sim2\%$ seen in Afshar’s wire grid). Measurement histograms and visibility curves from the simulator and device confirm the effectiveness of the DQS approach for optical interferometry with nontrivial Fock state encodings.

6. Broader Implications and Outlook

The digital simulation of complex bosonic interferometry and foundational quantum phenomena on universal quantum computers signals a qualitative expansion of DQS capabilities. With efficient mapping methods (e.g., Gray code) and carefully validated circuits, DQS now enables:

• Simulation of large-boson-number interferometry, quantum optics, and analogous field-theoretical processes using modest quantum resources
• Experimental paper of quantum foundational questions (e.g., complementarity) within well-controlled state-preparation and measurement frameworks
• Toolkit extensions for simulating quantum light–matter interactions, optical quantum computing circuits, or generalized quantum measurement protocols

This approach underscores both the computational efficiency and foundational significance of digital quantum simulation methods for bosonic systems and illustrates the role of quantum processors as experimentally relevant platforms for probing quantum optical and foundational phenomena (Brasil et al., 3 Feb 2025).

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Table: Gray code mapping and Pauli operator representation for bosonic simulation in DQS.

Fock State	Gray Code String	Representative Pauli Mapping
$|n\rangle$	$|G_1...G_{N_q}\rangle$	$$\prod_k \mathcal{P}_{G_k} \prod_{k'} \mathcal{Q}_{G_{k'}}$$

Efficient digital encoding and operator mapping are achieved due to the Gray code's one-bit-difference property, streamlining circuit construction for bosonic Hamiltonians in quantum simulation tasks.