Abstract Treating stabilizer operations as free, we establish lower bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks. Our bounds apply to adaptive computations using measurements with an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational subroutines including the quantum adder and the multiply-controlled Z gate. To prove our resource conversion bounds we introduce two new monotones, the stabilizer nullity and the dyadic monotone, and make use of the already-known stabilizer extent. We consider conversions that borrow resource states, known as catalyst states, and return them at the end of the algorithm. We show that catalysis is necessary for many conversions and introduce new catalytic conversions, some of which are optimal. By finding a canonical form for post-selected stabilizer computations, we show that approximating a single-qubit unitary to within diamond-norm precision ɛ requires at least 1/7 ⋅ log 2 (1/ ɛ ) − 4/3 T -states on average. This is the first lower bound that applies to synthesis protocols using fall-back, mixing techniques, and where the number of ancillas used can depend on ɛ . Up to multiplicative factors, we optimally lower bound the number of T or CCZ states needed to implement the ubiquitous modular adder and multiply-controlled- Z operations. When the probability of Pauli measurement outcomes is 1/2, some of our bounds become tight to within a small additive constant.
Abstract Treating stabilizer operations as free, we establish lower bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks. Our bounds apply to adaptive computations using measurements with an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational subroutines including the quantum adder and the multiply-controlled Z gate. To prove our resource conversion bounds we introduce two new monotones, the stabilizer nullity and the dyadic monotone, and make use of the already-known stabilizer extent. We consider conversions that borrow resource states, known as catalyst states, and return them at the end of the algorithm. We show that catalysis is necessary for many conversions and introduce new catalytic conversions, some of which are optimal. By finding a canonical form for post-selected stabilizer computations, we show that approximating a single-qubit unitary to within diamond-norm precision ɛ requires at least 1/7 ⋅ log 2 (1/ ɛ ) − 4/3 T -states on average. This is the first lower bound that applies to synthesis protocols using fall-back, mixing techniques, and where the number of ancillas used can depend on ɛ . Up to multiplicative factors, we optimally lower bound the number of T or CCZ states needed to implement the ubiquitous modular adder and multiply-controlled- Z operations. When the probability of Pauli measurement outcomes is 1/2, some of our bounds become tight to within a small additive constant.
In quantum computing, computational tasks are represented by quantum circuits. These circuits are composed of gates whose physical realization comes at a cost. Typically, gates from the so-called Clifford group are considered cheap, while non Clifford gates are considered expensive. Consequently, non-Clifford operations are often seen as a resource whose use should be minimized. In this thesis, following recent work by Beverland and others, we study lower bounds for the number of non-Clifford gates in quantum circuits. We focus on lower bounds that can be derived from monotones, which are real-valued functions of quantum states that are non-increasing under Clifford operations. We first provide a detailed presentation of two recently introduced monotones: the stabilizer nullity and the dyadic monotone. We then discuss how these monotones can be used to give lower bounds for the non-Clifford resources for two important quantum operations: the multiply-controlled Pauli Z gate and the modular adder.
Protecting quantum information from noise is essential for reliable quantum computation and quantum memories. Quantum state purification addresses this challenge by processing multiple noisy copies of an unknown quantum state to produce an output state of higher fidelity. Here we establish nearly tight upper and lower bounds on the non-Clifford resources required to implement the symmetric-projection purification protocol. For n copies of a d-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most ε using O(n) T gates, where the tilde-O notation suppresses polylogarithmic factors in n, d, and 1/ε. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error ε<211−1/n requires at least 2n−2 T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.
Motivated by their necessity for most fault-tolerant quantum computation schemes, we formulate a resource theory for magic states. First, we show that robustness of magic is a well-behaved magic monotone that operationally quantifies the classical simulation overhead for a Gottesman-Knill-type scheme using ancillary magic states. Our framework subsequently finds immediate application in the task of synthesizing non-Clifford gates using magic states. When magic states are interspersed with Clifford gates, Pauli measurements, and stabilizer ancillas-the most general synthesis scenario-then the class of synthesizable unitaries is hard to characterize. Our techniques can place nontrivial lower bounds on the number of magic states required for implementing a given target unitary. Guided by these results, we have found new and optimal examples of such synthesis.
We study the Clifford+T synthesis of quantum multiplexers U=⨁i=12nUi: an n-qubit control register selects which single-qubit gate Ui∈SU(2) acts on the target qubit. For each block Ui, consider the minimum even T-count needed to approximate it within diamond distance ε without ancillae, and let m be the largest of these costs. For an arbitrary multiplexer U, we construct an approximation of U within the same distance using at most m+O(2n(m+1))T gates. With one control qubit, the cost is m+O(1) and no ancillae are required. We also prove a lower bound that allows the Clifford+T circuit to use any number of clean ancillae initialized to zero and returned exactly to zero. For fixed n and independent Haar-random blocks, the minimum T-count among these circuits lies between (3−δ)log2(1/ε) and (3+δ)log2(1/ε) for every fixed δ>0, except with probability O(εcn,δ) as ε→0. The optimal leading coefficient is therefore the same as for a single Haar-random SU(2) gate. In the lower-bound proof, unitarity forces pairs of off-diagonal blocks describing transitions in opposite directions to satisfy cancellation relations with residuals of order ε2. We combine these relations with the arithmetic of Clifford+T circuits to bound the Haar measure of targets compatible with each pair of off-diagonal blocks. As an application, we obtain ancilla-free approximations of Haar-random SU(4) gates within diamond distance ε, using at most (9+δ)log2(1/ε)T gates for every fixed δ>0, except with probability O(εcδ) with cδ>0.
We introduce magic measures to quantify the nonstabilizerness of multiqubit quantum gates and establish lower bounds on the T count for fault-tolerant quantum computation. First, we introduce the stabilizer nullity of multiqubit unitary, which is based on the subgroup of the quotient Pauli group associated with the unitary. This unitary stabilizer nullity extends the state-stabilizer nullity by Beverland et al. [Quantum Sci. Technol. 5, 035009 (2020)] to a dynamic version. In particular, we show this nonstabilizerness measure has desirable properties, such as subadditivity under composition and additivity under tensor product. Second, we prove that a given unitary's stabilizer nullity is a lower bound for the T count, utilizing the above properties in gate synthesis. Third, we compare the state and the unitary stabilizer nullity, proving that the lower bounds for the T count obtained by the unitary stabilizer nullity are never less than the state-stabilizer nullity. Moreover, we show an explicit n-qubit unitary family of unitary stabilizer nullity 2n, which implies that its T count is at least 2n. This gives an example where the bounds derived by the unitary stabilizer nullity strictly outperform the state-stabilizer nullity by a factor of 2. We finally showcase the advantages of unitary stabilizer nullity in estimating the T count of quantum gates with interests.
Approximate synthesis of general single-qubit unitaries over the Clifford+T gate set
For the standard Clifford+T gate set, deterministic, ancilla-free synthesis now attains the minimal T-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T gate rotates by half the angle of T, generating a finer lattice of implementable operations. Access to this magic state is expected to lower the cost of deterministic and ancilla-free synthesis of general single-qubit unitaries (Kliuchnikov et al., arXiv:2203.10064), but no direct Clifford+T algorithm existed for this case. We provide one by extending the integer lattice-point enumeration method of Morisaki et al. We adopt a resource state cost model based on the magic-state catalysis approach of Gidney and Fowler (arXiv:1812.01238). On Haar-random targets synthesized to precisions ranging from ε=10−3 to 10−8, the cost of Clifford+T circuits scales as 2.4log2(1/ε) compared to 3.0log2(1/ε) for T-count-optimal Clifford+T circuits. Once a one-time catalyst state is amortized, the Clifford+T circuits are never costlier than their Clifford+T counterparts.