Toward Mixed Analog-Digital Quantum Signal Processing: Quantum AD/DA Conversion and the Fourier Transform

TL;DR

This work develops a new paradigm of mixed analog-digital QSP, providing a foundation for scalable analog-digital signal processing on quantum processors, and demonstrates how it naturally enables analog-digital conversion of quantum signals—specifically, the transfer of states between DV and CV quantum systems.

Abstract

Signal processing stands as a pillar of classical computation and modern information technology, applicable to both analog and digital signals. Recently, advancements in quantum information science have suggested that quantum signal processing (QSP) can enable more powerful signal processing capabilities. However, the developments in QSP have primarily leveraged \emph{digital} quantum resources, such as discrete-variable (DV) systems like qubits, rather than \emph{analog} quantum resources, such as continuous-variable (CV) systems like quantum oscillators. Consequently, there remains a gap in understanding how signal processing can be performed on hybrid CV-DV quantum computers. Here we address this gap by developing a new paradigm of mixed analog-digital QSP. We demonstrate the utility of this paradigm by showcasing how it naturally enables analog-digital conversion of quantum signals -- specifically, the transfer of states between DV and CV quantum systems. We then show that such quantum analog-digital conversion enables new implementations of quantum algorithms on CV-DV hardware. This is exemplified by realizing the quantum Fourier transform of a state encoded on qubits via the free-evolution of a quantum oscillator, albeit with a runtime exponential in the number of qubits due to information theoretic arguments. Collectively, this work marks a significant step forward in hybrid CV-DV quantum computation, providing a foundation for scalable analog-digital signal processing on quantum processors.

Figures (5)

• Figure 1: Schematic of the duality between (a) time-frequency-domain classical analog-digital signals versus (b) quantum position and momentum domain CV-DV signals. This work develops quantum AD/DA conversion protocols in panel (b) to facilitate mixed analog-digital quantum signal processing, in similar spirit to classical sampling/interpolation for mixed analog-digital classical signal processing in panel (a).
• Figure 2: (a): Illustration of the Gaussian basis states used in D/A conversion with single-variable QSP. The wave functions in position space are Gaussians of width $\sigma$, each separated by the spacing parameter $\Delta$. (b): The quantum circuit that implements D/A conversion with single variable QSP. Here, thin lines denote qubits, and the thick line an oscillator. Time proceeds left to right, enacting the gates depicted as boxes. The initial qubits state is $|\psi\rangle_Q$, and the initial oscillator state is $|0,\Delta\rangle_O^{\text{Gaus}}$. The first stage applies a series of controlled displacements $D( 2^{n-j} \Delta)$ between the qubits and oscillator. The second stage applies a series of operations that disentangle the qubits by flipping qubit $j$ conditioned on the bit $x_j$ of the oscillator's position. We depict these operations as an $X:= \sigma_x$ gate conditioned on $x_j$. In practice, each of these is realized as a QSP sequence $R_j(\hat{x})$ according to the construction of Eq. \ref{['eq:W_j_QSP']}. (c): The square wave functions $S_j(x\Delta)$ of Eq. \ref{['eq:square_wave_func']} for $j=1,2,3$, with $n=3$ and $\Delta=1$. Observe how at integer values $x$, these square waves are equal to $1-x_j$, which enables one to read out the bits $\{ x_j \}$.
• Figure 3: D/A conversion for various three-qubit states using $\textbf{(a)}$ single-variable QSP and $\textbf{(b)}$ non-Abelian QSP, including $\ket{\textrm{GHZ}}=(\ket{000}+\ket{111})/\sqrt{2}$ and $\ket{\textrm{W}}=(\ket{001}+\ket{010}+\ket{100})/\sqrt{3}$. $\textbf{(a)}$: We use a single-variable QSP sequence of degree $d=60$ with $\delta=0.2$, $\Delta=1$, and $\sigma = e^{-1.12} \approx 0.37$. As a metric for successful conversion, we estimate the purity (a measure of the degree to which the oscillator and qubits have been successfully disentangled) of the final oscillator state, yielding $0.976$, $0.958$, and $0.982$, respectively. These simulations were carried out in Bosonic Qiskit Biskit. $\textbf{(b)}$: We use non-Abelian QSP with $\Delta = \sqrt{2}$ and approximate the initial oscillator sinc state (defined in Eq. \ref{['eq:sinc_state']}) by a Gaussian with $\sigma = e^{-1.12} \approx 0.37$. The purities of the final oscillator state evaluate to $0.858$, $0.858$, and $0.858$, respectively. We used QuTiP johansson2012qutip for these simulations.
• Figure 4: (a): Illustration of two sinc basis states, as used in D/A conversion with non-Abelian QSP. The wave functions in position space are sinc functions (Eq. \ref{['eq:sinc_state']}), and have peaks that are each separated by the spacing parameter $\Delta$. (b): The circuit that implements D/A conversion with non-Abelian QSP, adapted from Ref. hastrup2022universal with the order of $W_n$ and $V_n$ flipped, which we believe to be a typo in Fig. 1 of Ref. hastrup2022universal. The initial qubits state is $|\psi\rangle_Q$, and the initial oscillator state is a sinc state $|0, \Delta \rangle_O^{\text{sinc}} = \frac{1}{\sqrt{\Delta}} \int dq \ \text{sinc}(\pi q /\Delta) |q\rangle_O$. Then, one applies a series of operations $V_j^\dag W_j^\dag$ between the oscillator and the $j^{\text{th}}$ qubit, where $V_j = e^{i \frac{\pi}{2^{j} \Delta} \hat{x}\hat{\sigma}_y^{(j)}}$ and $W_j = e^{ \pm i \frac{\Delta}{2} 2^{j-1}\hat{p}\hat{\sigma}_x^{(j)}}$. The systems on which these operations act are denoted by circles with dashed lines. In aggregate, this maps the initial qubits state to an equivalent oscillator state $|\psi\rangle_O$ encoded in a basis of displaced sinc states, as per Eq. \ref{['eq:Hastrup_inverse_state_transfer']}.
• Figure 5: The generic circuit used to implement the quantum Fourier transform of an $n$-qubit state $|\psi\rangle$ using AD/DA conversion. The initial state is first appended with ancilla qubits $|+\rangle^{\otimes a}$ to facilitate the QFT, as discussed in the main text. The unitaries $U_{\text{D/A}}(\Delta)$ are A/D and D/A conversion unitaries respectively (either single variable or non-Abelian), and $F$ is the Fourier gate of Eq. \ref{['eq:Fourier_gate']}. The circuit outputs the state $(U_{\text{QFT}}|\psi\rangle) |0\rangle^{\otimes a}$, from which the QFT may be obtained. For precise details on the construction of the QFT, see Supplemental Material Sec. SM.IV.

Theorems & Definitions (4)

• Theorem 1: Quantum AD/DA Conversion with Hybrid Single Variable QSP
• Theorem 2: Quantum AD/DA Conversion with Non-Abelian QSP
• Theorem 3: QFT from Oscillator Evolution and AD/DA Conversion (Single-Variable QSP
• Theorem 4: QFT from Oscillator Evolution and AD/DA Conversion Via Non-Abelian QSP