Quantum Frequential Computing: a quadratic runtime advantage for all computations

TL;DR

This work introduces quantum frequential computing, a paradigm where quantum control states, such as squeezed clock-like resources, drive gate applications at frequencies that scale quadratically with power relative to classical or semi-classical controls. By constructing both Hamiltonian and dynamical-semigroup models, the authors prove the existence of optimal conventional and quantum frequential computers and show that the quantum advantage can be realized with only a semi-classical internal bus. They analyze nonequilibrium steady-state operation, deriving linear power-to-frequency relations and bounding heat dissipation while highlighting the crucial role of renewal processes in maintaining stability. The results imply a universal, architecture-light pathway to large runtime speedups for any algorithm, with practical implications for power, cooling, and error management, and they outline concrete considerations for physical implementations using squeezed and standard quantum-limited oscillators. Overall, the paper provides a rigorous framework tying energy, power, and clocking to quantum-controlled gate dynamics, establishing a quadratic-in-power speedup that could significantly impact future scalable quantum architectures.

Abstract

An enduring challenge in computer science is reducing the runtime required to solve computational problems. Quantum computing has attracted significant attention due to its potential to deliver asymptotically faster solutions to certain problems compared to the best-known classical algorithms. This advantage is enabled by the quantum mechanical nature of the logical degrees of freedom. To date, it was unknown if permitting other parts of the computer to be quantum mechanical, rather than semi-classical, could yield additional runtime speed-ups as a function of resource utilization (e.g., power consumption or cooling requirements). In this work, we prove that when the control mechanisms associated with gate implementation are optimal quantum mechanical states, a quadratic runtime speedup (with respect to power consumption) is achievable for any algorithm, relative to optimal classical or semi-classical control schemes. Moreover, we demonstrate that only a small fraction of the computer's architecture needs to employ optimal quantum control states to realize this advantage, thereby significantly simplifying the design of future systems. We call this new device a quantum frequential computer, since the quantum speedup arises from an increase in gate frequency. In current state-of-the-art designs, gate frequency is often limited by the coupling strength between components. Notably, our approach achieves the speedup without requiring an increase in coupling strength.

Figures (5)

• Figure 1: a) Pictorial illustration of a conventional classical and quantum computer: the physical implementation of an algorithm realises a trajectory through physical state space of the information-bearing degrees of freedom (d.o.f.) and their control d.o.f.: This trajectory (dark blue line) passes through a sequence of "waypoints" (logical states) corresponding to the sequence of states dictated by the algorithm when compiled in machine code, i.e. the logical states generated by the logical gate set. Path-length corresponds to the algorithm's runtime. The difference between a classical and a quantum computer is that in the former the waypoints can only be classical logical states while in the latter they can be quantum. It is this "quantization of waypoints" along the trajectory which gives quantum computers their runtime advantage (for select problems), and why there is much excitement surrounding them. But this is only half the story: it is not only the total number of required waypoints which determines an algorithm's runtime, but also the time required to move the logical state between waypoints. The latter is determined by the nature of the gate control (e.g. "control pulses"). In quantum and classical computing, the gate control is considered to be a classical or semi-classical state. For a given energy in the control, this restricts the available trajectories of the logical state between two consecutive waypoints. FIG. 1. b) The same two sequences of waypoints as fig. 1 a) but now allowing for the control to be a quantum state: for the same energy in the gate control, the permissible trajectories between two consecutive waypoints is much larger and the path-length (now in orange) between consecutive waypoints, i.e. the time required to implement each gate, for the same given energy in the control, can be drastically reduced. This holds true regardless of whether the waypoints are purely classical or quantum. This constitutes a new type of computer, since the quantum runtime speedup holds for quantum and classical algorithms alike. We call it a quantum frequential computer.
• Figure 2: Diagram of the different systems involved: the memory $\textup{M}_0$, the logical space $\textup{L}$, and control $\textup{C}$. The logical space $\textup{L}$, is further divided into sub-register spaces, as in a conventional quantum or classical computation register (this substructure only enters indirectly via the to-be-defined-later values of $\tilde{d}({\mathbb{m}})$, $\mathbb{m}\in\mathcal{G}$).
• Figure 3: a) Schematic of the computer's architecture. The memory $\textup{M}$ and switches $\textup{W}$ are shown in their initial product states. The cells which the $k{^\text{th}}$ bus lane can read and write to are highlighted in blue while the cells in $\textup{M}_0$ on which logical gates $U(\mathbb{m})$, $\mathbb{m}\in\mathcal{G}$ are controlled on is depicted in orange. The $l{^\text{th}}$ bus lane can copy and write to memory cells in memory block $\textup{M}_{{\#},l}$ and can be turned on or off via changing the state of the switch in $\mathcal{C}_{\textup{W}_l}$ (which is located directly above the bus lane in the figure). During the $0{^\text{th}}$ cycle, the ${\bf 0}$ instructions in $\textup{M}_0$ are read, informing the control on $\textup{C}$ to turn the switches from off to on in a staggered fashion|No other change is present in $\textup{M}$, and no computation is performed. In the subsequent $l{^\text{th}}$ cycle, logical gates corresponding to the sequence $\mathbb{m}_{l,1}$, $\mathbb{m}_{l,2}$, $\ldots$, $\mathbb{m}_{l,N_g}$ is implemented hence performing computation. FIG. 3. b) A snapshot of the computer's memory at time $t\in [t_{k-1,l}, t_{k,l}]$, $l>0$ (subscript $x$ in $\mathbb{m}_{x,k}$ is mod $L$ and $\mathbb{m}_{0,k}:={\bf 0}$.). Memory cell $\textup{M}_{0,k}$ stores the appropriate gate sequence information for the gate being applied during time $[t_{k-1,l}, t_{k,l}]$, while other memory cells in other bus lanes (denoted by a question mark) are not necessarily in definite states in $\mathcal{C}_{\textup{M}_{{\#},l}}$ (the information is in transit between memory cells via the bus in preparation for later times).
• Figure 4: Qualitative illustration of representative dynamics of the oscillator on $\textup{C}$ in quadrature space over 3 cycles ($1{^\text{st}}$ cycle in blue, $2{^\text{nd}}$ in red, $3{^\text{rd}}$ in purple). Arrows indicate direction of dynamics over time. FIG. 4. a) State of the oscillator from \ref{['sec:quantum advantage']}: At the end of each cycle (of duration $T_0$) the oscillator gets close to the state it started in when initiating the cycle. In this way, small errors accumulate over many cycles ultimately leading to an intolerable error $E_R$. While we see from \ref{['thm:contrl with two clocks']} that said errors decrease rapidly with increasing power, for any given power, they eventually become large after sufficiently many cycles. FIG. 4. b) State of the oscillator from \ref{['sec:thm3 main text']}: With high probability, the oscillator is renewed to its initial state at the end of each cycle, leading to a quantum frequential computer in a nonequilibrium steady-state. Observe that the dynamics during each cycle are not identical, due to differing perturbations caused by the implementation of different gate sequences in each cycle. As such, this renewal cannot be unique: it must map many states to the same initial state, correcting for these small errors towards the end of each cycle to prevent them becoming large over many cycles. This many-to-one nature is a form of self-error correction. And, as we will see, it requires work to be done on it to run|similarly to Landauer's erasure principle or quantum error correction.
• Figure 5: Comparison between Landauer erasure and self-oscillation: two distinct physical processes with the commonality of requiring work to be done on the system by the environment to function due to the irreversibility of the process (the logical register in Landauer erasure, the self-oscillator in the computer). FIG. 5. a) The Landauer erasure process: an irreversible (many-to-one) channel is applied to three different register states to map them all to a unique register state $000$, thus erasing any information they contained. FIG. 5. b) the state of the self-oscillator (state on $\textup{C}$) during one cycle: at the end of the cycle's isentropic time interval, the state of the oscillator is a mixture over unknown perturbations to its orbit. At this stage, the self-oscillator enters an irreversible (non-isentropic) time interval, where work is done on the oscillator by the environment in order to restore it to its lower-entropy initial state, $\rho_\textup{C}^0$. This work $\braket{E^\textup{re}}$ is the power consumed by the oscillator per cycle.

Theorems & Definitions (56)

• Theorem 2.1: Upper bounds on computational speed
• Theorem 3.1: Optimal conventional and quantum frequential computers exist
• Theorem 4.1: Optimal quantum frequential computers only require a semi-classical internal bus
• Theorem 5.1: Nonequilibrium steady-state quantum frequential computers exist
• Theorem 6.1: Upper bounds on gate frequency in the nonequilibrium steady-state regime
• Lemma 6.0
• Lemma 6.0
• Theorem B.1: Upper bounds on computational speed
• proof
• Lemma C.1