[Paper Review] Efficient Inner-product Algorithm for Stabilizer States

Héctor J. García, Igor L. Markov+1 more|arXiv (Cornell University)|Oct 24, 2012

Quantum Computing Algorithms and Architecture18 references15 citations

TL;DR

This paper presents an efficient $O(n^3)$ algorithm for computing the inner product between stabilizer states, leveraging canonical circuit synthesis to map any stabilizer state to a computational basis state. The key contribution is proving that each $n$-qubit stabilizer state has exactly $4(2^n - 1)$ nearest-neighbor stabilizer states, verified via experiments, with extensions to stabilizer frames for arbitrary quantum states.

ABSTRACT

Large-scale quantum computation is likely to require massive quantum error correction (QEC). QEC codes and circuits are described via the stabilizer formalism, which represents stabilizer states by keeping track of the operators that preserve them. Such states are obtained by stabilizer circuits (consisting of CNOT, Hadamard and Phase only) and can be represented compactly on conventional computers using Omega(n^2) bits, where n is the number of qubits. Although techniques for the efficient simulation of stabilizer circuits have been studied extensively, techniques for efficient manipulation of stabilizer states are not currently available. To this end, we design new algorithms for: (i) obtaining canonical generators for stabilizer states, (ii) obtaining canonical stabilizer circuits, and (iii) computing the inner product between stabilizer states. Our inner-product algorithm takes O(n^3) time in general, but observes quadratic behavior for many practical instances relevant to QECC (e.g., GHZ states). We prove that each n-qubit stabilizer state has exactly 4(2^n - 1) nearest-neighbor stabilizer states, and verify this claim experimentally using our algorithms. We design techniques for representing arbitrary quantum states using stabilizer frames and generalize our algorithms to compute the inner product between two such frames.

Motivation & Objective

• To develop efficient algorithms for manipulating stabilizer states, which are central to quantum error correction and fault-tolerant quantum computation.
• To address the lack of efficient techniques for computing inner products between stabilizer states despite known efficient simulation of stabilizer circuits.
• To design a canonical circuit synthesis algorithm that maps any stabilizer state to a computational basis state using a structured $H$-$C$-$CZ$-$P$-$H$ block sequence.
• To prove and verify the exact number of nearest-neighbor stabilizer states for any $n$-qubit stabilizer state, a geometric property of interest in quantum state space.

Proposed method

• Leverage theoretical insights from prior work to design a canonical circuit synthesis algorithm that transforms any stabilizer state into a computational basis state using a $H$-$C$-$CZ$-$P$-$H$ block-structured circuit.
• Use the stabilizer formalism to represent stabilizer states via generators of the stabilizer group, enabling compact $O(n^2)$ classical representation.
• Implement an inner product algorithm that computes $|raket{\psi|\varphi}|$ by synthesizing a circuit mapping $|\psi\rangle$ to $|0\cdots0\rangle$, then measuring the overlap with $|\varphi\rangle$.
• Generalize the framework to stabilizer frames—representations of arbitrary quantum states using stabilizer states and phase factors—to extend inner product computation beyond pure stabilizer states.
• Apply the algorithm to verify the geometric claim that each $n$-qubit stabilizer state has exactly $4(2^n - 1)$ nearest-neighbor stabilizer states, defined as those with maximal $|\braket{\psi|\varphi}| \neq 1$.
• Use experimental evaluation on quantum codes and entangled states (e.g., GHZ states) to demonstrate quadratic performance in practice despite $O(n^3)$ worst-case complexity.

Experimental results

Research questions

• RQ1 What is the exact number of nearest-neighbor stabilizer states for any $n$-qubit stabilizer state, and can this be proven and verified computationally?
• RQ2 Can an efficient algorithm be designed to compute the inner product between two stabilizer states using canonical circuit synthesis?
• RQ3 How can the stabilizer formalism be extended to represent arbitrary quantum states, and what is the inner product between such generalized frames?
• RQ4 What is the practical performance of the inner product algorithm on relevant quantum information states such as GHZ and error-correcting code states?
• RQ5 Can the canonical circuit synthesis algorithm be constructed efficiently to enable fast state transformation and inner product computation?

Key findings

• The paper proves that each $n$-qubit stabilizer state has exactly $4(2^n - 1)$ nearest-neighbor stabilizer states, defined as those with maximal nontrivial inner product magnitude.
• The inner product algorithm runs in $O(n^3)$ time in general, but exhibits quadratic scaling for practical instances such as GHZ states and quantum error-correcting code states.
• The algorithm successfully verifies the theoretical count of nearest neighbors through experimental evaluation on various stabilizer states.
• The framework is generalized to stabilizer frames, enabling the computation of inner products between arbitrary quantum states represented as superpositions of stabilizer states.
• The canonical circuit synthesis algorithm produces a $H$-$C$-$CZ$-$P$-$H$ block-structured circuit for any input stabilizer state, enabling efficient state transformation and overlap computation.
• Performance evaluation shows that the algorithm is efficient in practice, with runtime scaling as $O(n^2)$ for many relevant quantum states, including entangled and code states.