Quantum state preparation and block encoding are versatile and practical input models for quantum algorithms in scientific computing. The circuit complexity of state preparation and block encoding frequently dominates the end-to-end gate complexity of quantum algorithms. We give algorithms with lower C-NOT counts for both the state preparation and block encoding. For a general $n$-qubit state, we improve the C-NOT count of the Plesch-Brukner algorithm (2011) from $(23/24)\times 2^n$ to $(11/12)\times2^n$. For block encoding, our single-ancilla protocol for $2^{n-1}\times 2^{n-1}$ matrices uses the spectral norm as subnormalization and achieves a C-NOT count leading term $(11/48)\times 4^n$. Further optimization is performed for low-rank matrices, which frequently arise in practical applications. Specifically, we achieve the C-NOT count leading term $(2^{\lceil\log_{2}K\rceil}+(11/12))\times 2^n$ for a rank-$K$ matrix. This is the first quantum algorithm that encodes matrices using the optimal normalization factor while also allowing the C-NOT count to be adjusted according to the matrix rank. Our approach builds upon the recursive Block-ZXZ decomposition from Krol et al. (2024) and introduces a diagonal matrix migration technique based on the commutativity of the diagonal matrix and the uniformly controlled rotation about the $z$-axis to minimize the use of C-NOT gates.
Quantum state preparation and block encoding are versatile and practical input models for quantum algorithms in scientific computing. The circuit complexity of state preparation and block encoding frequently dominates the end-to-end gate complexity of quantum algorithms. We give algorithms with lower C-NOT counts for both the state preparation and block encoding. For a general n-qubit state, we improve the C-NOT count of the Plesch-Brukner algorithm (2011) from (23/24)×2n to (11/12)×2n. For block encoding, our single-ancilla protocol for 2n−1×2n−1 matrices uses the spectral norm as subnormalization and achieves a C-NOT count leading term (11/48)×4n. Further optimization is performed for low-rank matrices, which frequently arise in practical applications. Specifically, we achieve the C-NOT count leading term (2⌈log2K⌉+(11/12))×2n for a rank-K matrix. This is the first quantum algorithm that encodes matrices using the optimal normalization factor while also allowing the C-NOT count to be adjusted according to the matrix rank. Our approach builds upon the recursive Block-ZXZ decomposition from Krol et al. (2024) and introduces a diagonal matrix migration technique based on the commutativity of the diagonal matrix and the uniformly controlled rotation about the z-axis to minimize the use of C-NOT gates.
Block-encoding operators are one of the essential components in quantum algorithms based on Quantum Signal Processing. Their gate complexity largely determines the overall gate complexity of the full algorithm. Using variational methods, we compile single-ancilla block-encoding unitaries with near-optimal resource requirements for a large range of input matrices. We find that the number of variational parameters in the parameterized quantum circuit approaches the number of free parameters in the input matrices, depending on whether they are real, complex and/or hermitian. Additionally, symmetries present in the input matrix can be incorporated into the ansatz circuit, reducing the parameter count further and enhancing optimizability. While performing a variational compilation of block-encodings ceases to be computationally feasible for large system sizes, the constructed operators can be used as components of larger block-encodings via a linear combination of block-encodings.
Many quantum algorithms require coherent access to classical data, often modeled by quantum read-only memory (QROM). We initiate the study of the T count of sparse QROM, in which only s of the 2n addresses store nonzero data. We prove that the optimal T count is Θ(n+min{s,s(m+log(2n+1/s))}). Our upper bounds use a multilevel hashing scheme, while our lower bounds reduce sparse QROM to state preparation and use counting arguments for adaptive Clifford+T circuits. The lower bounds thus hold even when mid-circuit measurements and classically controlled operations are allowed. As applications, we obtain matching T-count bounds Θ(min{s,slog(2n+1/s)}+slog(1/ε)+log(1/ε)) for s-sparse state preparation and Θ(2ns(n+log(1/εBE))+log(1/εBE)) for block encoding of s-sparse matrices, where ε and εBE are the precision of state preparation and block encoding, respectively.
Quantum State Preparation Algorithms consist of defining a sequence or unitary operations to load a specific target state on a quantum computer. We can use those algorithms in appli- cations such as quantum machine learning. However, some state preparation algorithms have exponential circuit complexity with the number of qubits on the system. That is the case of amplitude encoding algorithms, which is an encoding type for loading normalized data into the probability amplitudes of the state. To circumvent this overhead in circuits’ complexity, works explore specific properties of quantum states to optimize the circuit’s complexity, such as sparsity or symmetry. Other works explore simplifying the quantum circuit to load an ap- proximate quantum state. It is the case of Quantum Generative Adversarial Networks, which use a specific circuit architecture comprised of alternating blocks of single-qubit rotations and two-qubit entangling controlled gates. But when trained to load random distributions on, we observed the performance deteriorates as the number of qubits increases in terms of relative entropy. In this work, we propose different architectures for the Quantum Generative mod- els based on the state preparation algorithm known as Low-Rank. Through experiments for loading the log-normal distribution, we show error reductions in quantum state initialization.
We determine the non-Clifford T-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford+T model, when arbitrarily many clean ancillas and unrestricted block-encoding subnormalization are allowed, but without mid-circuit measurements or classical feed-forward. Our main technical tool is an ancilla-compression theorem: any block encoding of an n-qubit operator with a clean ancillas and at most sT gates can be compressed to use at most min{a,n+2s} ancillas, without increasing the absolute error or T-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error ε, a volume-covering argument combined with circuit counting gives the worst-case lower bound Ω(n2log(n4/ε)), matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on n spins, we obtain independent lower bounds Ω(n) from stabilizer nullity and Ω(log(1/ε)) from one-qubit state preparation, established using different Hamiltonian instances. Together with an explicit LCU construction, they give the tight worst-case scaling Θ(n+log(1/ε)). As an application, we evaluate the T-count of a Hamiltonian simulation circuit based on quantum singular value transformation, with each block-encoding query compiled separately. When phase synthesis and controlled queries add at most constant-factor overhead, the simulation T-count scales as the query count times the optimal T-count per query.
The problems of quantum state preparation and matrix block-encoding are ubiquitous in quantum computing: they are crucial parts of various quantum algorithms for the purpose for initial state preparation as well as loading problem relevant data. We first present an algorithm based on QRS that prepares a quantum state ∣ψf⟩∝∑x=1Nf(x)∣x⟩. When combined with efficient reference states the algorithm reduces the cost of quantum state preparation substantially, if certain criteria on f are met. When the preparation of the reference state is not the dominant cost, and the function f and relevant properties are efficiently computable or provided otherwise with cost o(N), the QRS-based method outperforms the generic state preparation algorithm, which has cost O(N). We demonstrate the detailed performance (in terms of the number of Toffoli gates) of the QRS-based algorithm for quantum states commonly appearing in quantum applications, e.g., those with coefficients that obey power law decay, Gaussian, and hyperbolic tangent, and compare it with other methods. Then, we adapt QRS techniques to the matrix block-encoding problem and introduce a QRS-based algorithm for block-encoding a given matrix A=∑ijAij∣i⟩⟨j∣. We work out rescaling factors for different access models, which encode how the information about the matrix is provided to the quantum computer. We exemplify these results for a particular Toeplitz matrix with elements Aij=1/∥i−j∥2, which appears in quantum chemistry, and PDE applications, e.g., when the Coulomb interaction is involved. Our work unifies, and in certain ways goes beyond, various quantum state preparation and matrix block-encoding methods in the literature, and gives detailed performance analysis of important examples that appear in quantum applications.
Quantum state preparation is a central primitive in many quantum algorithms, yet it is generally resource intensive, with efficient constructions known only for structured families of states. This work introduces a method for preparing quantum states whose amplitudes are given by a degree−d polynomial, using circuits with logarithmic depth in the number n of qubits and only O(n) ancilla qubits, improving previous approaches that required linear-depth circuits. The construction first relies on a block-encoding of an affine diagonal operator based on its Pauli-basis decomposition, which involves only n terms. A modified linear-combination-of-unitaries (LCU) technique is introduced to implement this decomposition in logarithmic depth, together with a novel circuit for the EXACT-one oracle that flags basis states in which exactly one qubit is in the state ∣1⟩. It then uses a generalized quantum eigenvalue transformation (GQET) to promote this affine operator to an arbitrary degree polynomial. Theoretical analysis and numerical simulations are reported along with a proof-of-principle implementation on a trapped-ion quantum processor using 14 qubits and more than 500 primitive quantum gates. Because polynomial approximations are ubiquitous in scientific computing, this construction provides a scalable and resource-efficient approach to quantum state preparation, further improving the potential of quantum algorithms in fields such as chemistry, physics, engineering, and finance.