Harrow-Hassidim-Lloyd algorithm (HHL) allows for the exponentially faster solution of a system of linear equations. However, this algorithm requires the postselection of an ancilla qubit to obtain the solution. This postselection makes the algorithm result probabilistic. Here we show conditions when the HHL algorithm can work without postselection of the ancilla qubit. We derive expectation values for an observable $M$ on the HHL outcome state when ancilla qubit is measured in $|0\ensuremath{\rangle}$ and $|1\ensuremath{\rangle}$ and show a condition for postselection-free HHL running. We provide an explicit example of a practically interesting input matrix and an observable, which satisfies the postselection-free HHL condition. Our work can improve the performance of the HHL-based algorithms.
Harrow-Hassidim-Lloyd algorithm (HHL) allows for the exponentially faster solution of a system of linear equations. However, this algorithm requires the postselection of an ancilla qubit to obtain the solution. This postselection makes the algorithm result probabilistic. Here we show conditions when the HHL algorithm can work without postselection of the ancilla qubit. We derive expectation values for an observable M on the HHL outcome state when ancilla qubit is measured in ∣0\ensuremath⟩ and ∣1\ensuremath⟩ and show a condition for postselection-free HHL running. We provide an explicit example of a practically interesting input matrix and an observable, which satisfies the postselection-free HHL condition. Our work can improve the performance of the HHL-based algorithms.
We propose an iterative improvement method for the Harrow-Hassidim-Lloyd (HHL) algorithm to solve a linear system of equations. This is a quantum-classical hybrid algorithm. The accuracy is essential to solve the linear system of equations. However, the accuracy of the HHL algorithm is limited by the number of quantum bits used to express the eigenvalues of the matrix. Our iterative method improves the accuracy of the HHL solutions, and gives higher accuracy which surpasses the accuracy limited by the number of quantum bits. In practical HHL algorithm, a huge number of measurements is required to obtain good accuracy, even if we provide a sufficient number of quantum bits for the eigenvalue expression, since the solution is statistically processed from the measurements. Our improved iterative method can reduce the number of measurements. Moreover, the sign information for each eigenstate of the solution is lost once the measurement is made, although the sign is significant. Therefore, the naïve iterative method of the HHL algorithm may slow down, especially, when the solution includes wrong signs. In this paper, we propose and evaluate an improved iterative method for the HHL algorithm that is robust against the sign information loss, in terms of the number of iterations and the computational accuracy.
We extend the Harrow-Hassidim-Lloyd (HHL) algorithm, which is well-studied in the qubit framework, to its qutrit counterpart (which we call qutrit HHL, as opposed to qubit HHL, which is HHL using qubits), and develop a program for its implementation. We design Weyl-Heisenberg gadgets, the qutrit equivalents of Pauli gadgets, and come up with a practical implementation scheme for qutrit HHL. We test HHL with qutrits for simple matrices and verify the results against the expected outcomes. We apply the algorithm to quantum chemistry, and in particular, to the potential energy curve calculations of the model problem of the Hydrogen molecule in the split valence basis. We do so for two cases: 1-qutrit and 2-qutrit input states, where the latter makes use of our gadgets. We compare the number of qudits and the number of gates required between qubit and qutrit HHL implementations. In general, we find that for a fixed precision, the qutrit HHL circuit requires fewer number of qudits and comparable number of two-qudit gates than its qubit counterpart.
The Harrow-Hassidim-Lloyd (HHL) algorithm offers exponential speedup for solving the quantum linear-system problem. But some caveats for the speedup could be hard to met. One of the difficulties is the encoding bottleneck, i.e., the efficient preparation of the initial quantum state. To prepare an arbitrary N-dimensional state exactly, existing state-preparation approaches generally require a runtime of O(N), which will ruin the speedup of the HHL algorithm. Here we show that the states can be prepared approximately with a runtime of O(poly(logN)) by employing a slightly modified version of the HHL algorithm itself. Thus, applying this approach to prepare the initial state of the original HHL algorithm can preserve the exponential speedup advantage. It can also serve as a standalone solution for other applications demanding fast state preparation.
Although the Harrow-Hassidim-Lloyd (HHL) algorithm offers an exponential speedup in system size for treating linear equations of the form A x ⃗ = b ⃗ on quantum computers when compared to their traditional counterparts, it faces a challenge related to the condition number ( κ ) scaling of the A matrix. In this work, we address the issue by introducing the postselection-improved HHL (Psi-HHL) framework that operates on a simple yet effective premise: subtracting mixed and wrong signals to extract correct signals while providing the benefit of optimal scaling in the condition number of A (denoted as κ ) for large κ scenarios. This approach, which leads to minimal increase in circuit depth, has the important practical implication of having to use substantially fewer shots relative to the traditional HHL algorithm. The term “signal” refers to a feature of | x 〉 . We design circuits for overlap and expectation value estimation in the Psi-HHL framework. We demonstrate performance of Psi-HHL via numerical simulations. We carry out two sets of computations, where we go up to 26-qubit calculations, to demonstrate the ability of Psi-HHL to handle situations involving large- κ matrices via (a) a set of toy matrices, for which we go up to size 64 × 64 and κ values of up to ≈ 1 × 10 6 , and (b) application to quantum chemistry, where we consider matrices up to size 256 × 256 that reach κ of about 393. The molecular systems that we consider are Li 2 , KH, RbH, and CsH.
The Harrow-Hassidim-Lloyd (HHL) algorithm is a quantum algorithm for solving systems of linear equations that, in principle, offers an exponential improvement in scaling with the system size compared to classical approaches. In this work, we investigate the practical implementation and optimisation of the HHL algorithm with a focus on improving its performance on near-term quantum simulators. After outlining the algorithm, we examine two optimisation strategies aimed at improving fidelity and scalability: Suzuki-Trotter decomposition of the Hamiltonian evolution operator and a block-encoding approach that embeds the problem matrix into a larger unitary operator. The performance of these methods is evaluated through simulations on matrices with varying sparsity, including diagonal, tridiagonal, moderately dense, and fully dense cases. Our results show that while HHL achieves near-ideal fidelity for highly structured matrices, performance degrades as sparsity decreases due to the increasing cost of Hamiltonian simulation and reduced post-selection probability due to higher condition number. Block encoding is found to provide improved fidelity for moderately dense matrices, whereas Trotterisation offers a qubit-efficient approach for sparse systems. These results highlight the importance of matrix structure in determining the practical efficiency of HHL and inform future implementations that combine algorithmic optimisation with hardware-aware design.
Solving systems of linear equations is important both in classical and quantum computing. Harrow-Hassidim-Lloyd algorithm (HHL algorithm), a quantum algorithm for solving systems of linear equations, has achieved exponentially improved performance compared with corresponding classical algorithm. We will show in this article that the HHL algorithm is actually a duality quantum algorithm with non- unitary transformation from initial to the final state. We present a detailed realization of the HHL algorithm in a duality quantum computing formalism that allows the construction of non- unitary operations. The divider structure, combinor structure, and the total quantum circuit for the HHL algorithm in the duality quantum computing formalism are given explicitly.