A Temperature Change can Solve the Deutsch-Jozsa Problem : An Exploration of Thermodynamic Query Complexity

TL;DR

A thermodynamic model of quantum query complexity is introduced, showing how qubit thermal machines can act as oracles, queried via heat exchange with a probe, providing a novel thermodynamic solution to the Deutsch-Jozsa problem.

Abstract

We demonstrate how a single heat exchange between a probe thermal qubit and multi-qubit thermal machine encoding a Boolean function, can determine whether the function is balanced or constant, thus providing a novel thermodynamic solution to the Deutsch-Jozsa problem. We introduce a thermodynamic model of quantum query complexity, showing how qubit thermal machines can act as oracles, queried via heat exchange with a probe. While the Deutsch-Jozsa problem requires an exponential encoding in the number of oracle bits, we also explore a restricted Bernstein-Vazirani problem, which admits a linear thermal oracle and a single thermal query solution. We establish bounds on the number of samples needed to determine the probe temperature encoding the solution for the Deutsch-Jozsa problem, showing that it remains constant with problem size. Additionally, we propose a proof-of-principle experimental implementation to solve the 3-bit Bernstein-Vazirani problem via thermal kickback. This work bridges thermodynamics and complexity theory, suggesting that quantum thermodynamics could provide an unconventional route to computing beyond classical computation.

Figures (7)

• Figure 1: (i) A unitary binary oracle takes as input an $n$-qubit string $\ket{x} \,:\, x\in \{0,1\}^{\times n}$ and a single ancilla qubit $\ket{a} \, : \, a \in \{0,1\}$ imprinting the output of the Boolean function encoded by the oracle into $U_f$ onto the ancilla as $\ket{a \oplus f(x)}$. (ii) A thermal machine oracle $\tau_f$ is a collection of qubits whose energetic structure encodes $f$. An agent with access to a probe$\rho_{S}(\beta)$ can exchange heat with $\tau_f$ via $U_x$ depending on an input $n$-bitstring $x$. The probe state changes to $\rho_S(\beta'_x)$ where the temperature of $S$ encodes the output $f(x)$.
• Figure 2: Three circuit diagrams depicting different heat exchanges across the probe qubit and thermal machine oracle which result in three different queries which are examined below for the Deutsch-Jozsa problem.
• Figure 3: A visualisation of an illustrative example. At the left, we see the standard 3 qubit Deutsch-Jozsa circuit where an oracle of the 2-bit function $f(x)$ is implemented as a unitary and the decision problem is solved by evalutating $\braket{00|\psi}$. To the right, we see a 4 qubit thermal machine at temp. $T_M$ with gaps $\Gamma = (\gamma_1, \gamma_2, \gamma_3, \gamma_4)$ whose energy level structure is used by an oracle to encode $f(x)$, global properties of $f(x)$ can be determined in a single heat exchange with a probe at temp. $T_S$ with energetic gap $\omega$.
• Figure 4: A plot showing the inv. temp. of the probe with $\omega = 1$ after thermal query $\beta'_S$ against the initial temp. $\beta_S$ of the probe for the 2-bit DJ problem. We see that the different outcomes are less distinguishable as the thermal oracle qubits become warmer i.e. smaller $\beta_M$ (solid lines).
• Figure 5: A plot showing the L.H.S of inequality eq.\ref{['eq:dist_cond']}$(1 + e^{-\beta_M E_1})^{-N} - (1 + e^{-\beta_M E_1})^{-N/2}(1 + e^{-\beta_M E_2})^{-N/2}$ for $\beta_M = 1$ and various oracle qubit energies $E_1$ and $E_2$, showing that $t \in [0,0.5]$ is achievable with different energies for different $n$.