Quantum phase estimation with optimal confidence interval using three control qubits

TL;DR

The paper addresses the challenge of obtaining optimal confidence intervals in quantum phase estimation (QPE) with limited use of the unitary $U$, by employing discrete prolate spheroidal sequence (DPSS) states as the control state. It proposes an efficient DPSS state preparation based on a matrix product state (MPS) with bond dimension $\chi=4$, enabling scalable, real-time DPSS preparation and compatibility with the semi-classical QFT. The results show near-perfect DPSS fidelity for dimensions up to $D$ corresponding to $2^{24}$ and demonstrate that DPSS provides the best possible CI width at a fixed confidence level, with only minor degradation when using an $\chi=4$ MPS approximation; in power-of-two dimensions, QPE can be carried out with just three control qubits. The work also analyzes implementation costs, showing reduced non-Clifford gate counts and fewer ancilla qubits compared to prior DPSS state-preparation methods, making DPSS-enabled QPE more practical for near-term fault-tolerant quantum computers. Overall, the approach offers a scalable, high-fidelity route to optimal CI-based phase estimation across DPSS dimensions while highlighting the trade-offs between dimension, fidelity, and resource costs.

Abstract

Quantum phase estimation is an important routine in many quantum algorithms, particularly for estimating the ground state energy in quantum chemistry simulations. This estimation involves applying powers of a unitary to the ground state, controlled by an auxiliary state prepared on a control register. In many applications the goal is to provide a confidence interval for the phase estimate, and optimal performance is provided by a discrete prolate spheroidal sequence. We show how to prepare the corresponding state in a far more efficient way than prior work. We find that a matrix product state representation with a bond dimension of 4 is sufficient to give a highly accurate approximation for all dimensions tested, up to $2^{24}$. This matrix product state can be efficiently prepared using a sequence of simple three-qubit operations. When the dimension is a power of 2, the phase estimation can be performed with only three qubits for the control register, making it suitable for early-generation fault-tolerant quantum computers with a limited number of logical qubits.

Figures (11)

• Figure 1: Variant of the first steps of the QPE algorithm with $D-1$ applications of controlled unitary operations, where $D$ may be any positive integer.
• Figure 2: Diagram (a) illustrates an operator $M^{[k]}$, which can be constructed by reordering the rows and columns of $V^{[k]}$, shown in diagram (b). The bottommost qubit is the most significant qubit. The slashes on the lines indicate that the number of qubits needed varies according to the dimension needed to span the corresponding index. Diagram (c) shows how an operator $M^{[k]}$ connects to neighbouring operators in an MPS. For $\chi=4$, most operators in the MPS will have two qubits indexing $\alpha_{k-1}$ and $\alpha_k$, and one qubit in the state $\ket{0}$ needed to match the dimension on either side.
• Figure 3: Quantum circuit for preparing an MPS with $\chi=4$. All operators between $M^{[2]}$ and $M^{[n-2]}$ act on three consecutive qubits. The bottommost qubit is the most significant qubit.
• Figure 4: Circuit for performing QPE for a unitary operator $U$ and its eigenstate $\ket{\phi}$ using the semi-classical Fourier transform. The state on the control register is an MPS with $\chi=4$, requiring no more than 3 qubits at any given time.
• Figure 5: Circuit for preparing a $D$-dimensional DPSS state on $n$ qubits. The $-P$ operator subtracts $P$ from the register and the $\geq D$ operator flips the ancilla qubit if the value in the bottom register is greater than or equal to $D$. The preparation succeeds if the state $\ket{0}$ is measured on the ancilla qubit.