We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored $T$ gate magic state. This lowers its asymptotic bound to $2^{\sim 0.463 t}$ for multi-Pauli measurements on $t$ magic states, improving over the best previously found bound of $2^{\sim 0.468 t}$. We numerically demonstrate this reduction. This constructively produces the most efficient strong simulation algorithm of the Clifford+$T$ gateset to relative or multiplicative error. We then examine the cost of Pauli measurement in terms of its Gauss sum rank, which is a slight generalization of the stabilizer rank and is a lower bound on its asymptotic scaling. We demonstrate that this lower bound appears to be tight at low $t$-counts, which suggests that the stabilizer rank found at the twelve-qubit state can be lowered further to $2^{\sim 0.449 t}$ and we prove and numerically show that this is the case for single-Pauli measurements. Our construction directly shows how the reduction at $12$ qubits is iteratively based on the reduction obtained at $6$, $3$, $2$, and $1$ qubits. This explains why novel reductions are found at tensor factors for these number of qubit primitives, an explanation lacking previously in the literature. Furthermore, in the process we observe an interesting relationship between the T gate magic state's stabilizer rank and decompositions that are Clifford-isomorphic to a computational sub-basis tensored with single-qubit states that produce minimal unique stabilizer state inner products -- the same relationship that allowed for finding minimal numbers of unique Gauss sums in the odd-dimensional qudit Wigner formulation of Pauli measurements.
We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored T gate magic state. This lowers its asymptotic bound to 2∼0.463t for multi-Pauli measurements on t magic states, improving over the best previously found bound of 2∼0.468t. We numerically demonstrate this reduction. This constructively produces the most efficient strong simulation algorithm of the Clifford+T gateset to relative or multiplicative error. We then examine the cost of Pauli measurement in terms of its Gauss sum rank, which is a slight generalization of the stabilizer rank and is a lower bound on its asymptotic scaling. We demonstrate that this lower bound appears to be tight at low t-counts, which suggests that the stabilizer rank found at the twelve-qubit state can be lowered further to 2∼0.449t and we prove and numerically show that this is the case for single-Pauli measurements. Our construction directly shows how the reduction at 12 qubits is iteratively based on the reduction obtained at 6, 3, 2, and 1 qubits. This explains why novel reductions are found at tensor factors for these number of qubit primitives, an explanation lacking previously in the literature. Furthermore, in the process we observe an interesting relationship between the T gate magic state's stabilizer rank and decompositions that are Clifford-isomorphic to a computational sub-basis tensored with single-qubit states that produce minimal unique stabilizer state inner products -- the same relationship that allowed for finding minimal numbers of unique Gauss sums in the odd-dimensional qudit Wigner formulation of Pauli measurements.
Bravyi and Gosset [S. Bravyi and D. Gosset, Phys. Rev. Lett. 116, 250501 (2016)] recently gave classical simulation algorithms for quantum circuits dominated by Clifford operations. These algorithms scale exponentially with the number of T gates in the circuit, but polynomially in the number of qubits and Clifford operations. Here we extend their algorithm to qudits of odd prime dimension. We generalize their approximate stabilizer rank method for weak simulation to qudits and obtain the scaling of the approximate stabilizer rank with the number of single-qudit magic states. We also relate the canonical form of qudit stabilizer states to Gauss sum evaluations and give an O(n3) algorithm for calculating the inner product of two n-qudit stabilizer states.
In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of m copies of the magic state | T ⟩ = 2 − 1 ( | 0 ⟩ + e i π / 4 | 1 ⟩ ) in the limit of large m . In particular, we show that | T ⟩ ⊗ m can be exactly expressed as a superposition of at most O ( 2 α m ) stabilizer states, where α ≤ 0.3963 , improving on the best previously known bound α ≤ 0.463 . This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an n -qubit Clifford + T circuit U with m uses of the T gate to within a given inverse polynomial relative error using a runtime p o l y ( n , m ) 2 α m . We also provide improved upper bounds on the stabilizer rank of symmetric product states | ψ ⟩ ⊗ m more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and m instances of any (fixed) single-qubit Z -rotation gate with runtime poly ( n , m ) 2 m / 2 . We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.
One of the state of the art techniques for classically simulating quantum circuits relies on approximating the output state of the circuit by a superposition of stabilizer states. \nIf the number of non-Clifford gates in the circuit is small, such simulations can be very effective. \nThis thesis provides various improvements in this framework. First, we describe an improved method of computing approximate stabilizer decompositions, which reduces the time cost of computing a single term in the decomposition from O(elln2) to O(mn2), where ell is the total number of gates in the circuit, and m is the number of non-Clifford gates. \nSince this subroutine has to be repeated exponentially many times, this improvement can be significant in practice whenever ellggm. \nOur method uses a certain re-writing of the circuit, which in some cases allows for a significant amelioration of the exponential scaling of the required classical resources. \n \nFurthermore, we describe a method of constructing exact, low-rank stabilizer decompositions of ketpsiotimesm, where ketpsi is either a magic state or an equatorial state. \nFor any single qubit magic state ketpsi, we find stabilizer decompositions of ketpsiotimesm with 2mlog2(3)/4 terms. This improves on the best known bound of 2mlog2(7)/6. \nSimilarly, for any single qubit equatorial state ketpsi, we give a stabilizer decomposition of ketpsiotimesm with 2m/2 terms. \nTo our knowledge no such decompositions were previously known. \nThese results translate to milder exponential scaling of the classical resources required for estimating probabilities of quantum circuits up to a polynomially small multiplicative error, as well as allowing more types of circuits to be simulated in this way. \n \nWe also consider certain obstructions to classical simulations. \nIt has been argued in various contexts that contextuality and non-locality hamper classical simulations of quantum circuits. \nLinear constraint systems (LCSs) are a generalization of the well-known Peres-Mermin magic square, which has been recently used to prove a separation between the power of constant depth classical and quantum circuits. \n \nWhile binary LCSs have been studied in detail, d-ary LCSs are less well-understood. In this thesis we consider linear constraint systems modulo d>2. We give a simple proof, of the previously known fact, that any linear constraint system which admits a quantum solution consisting of generalized Pauli observables in odd dimension must be classically satisfiable. We further prove that, for odd d, if a Pauli-like commutation relation between two variables in the LCS arises, then it has no quantum solutions in any dimensions, in stark contrast to the even d case. We apply this result to various examples, for instance showing that many generalizations of the Peres-Mermin magic square do not give rise to a quantum vs. classical satisfiability gap.
Distinct Clifford orbits of magic states can exhibit different stabilizer ranks at small tensor powers. We establish this for qutrits, where the single-qutrit Clifford group has four inequivalent orbits of magic states: Strange, Norrell, Hadamard-eigenstate, and the qutrit T-state, but a nontrivial upper bound on the asymptotic exponent had been pinned down for only the qutrit T-state. For the other three orbits we give explicit stabilizer decompositions, yielding upper bounds on the per-copy asymptotic stabilizer-rank exponent: γS≤log3(2)/2≈0.316 for the Strange state, and γH3,γN≤log3(4)/3≈0.421 for the Hadamard-eigenstate and Norrell orbits, all strictly below the prior γT3≤1/2 baseline. We also prove the first nontrivial Ω(m/logm) asymptotic lower bounds for the Hadamard-eigenstate and Norrell orbits, and exhibit two-qutrit Clifford circuits that convert two copies of these states into an injectable phase state with constant success probability, enabling constant-overhead injection of one non-Clifford diagonal gate per orbit. In the case of qubits, we give a closed-form decomposition of the qubit T-type orbit at four copies matching the existing γT≤log2(3)/4≈0.396 exponent via a direct algebraic identity rather than an entangled cat-state construction. An open-source library stabrank accompanies the paper, with Lean 4 proof formalizations of all the decompositions.
Improved Weak Simulation of Universal Quantum Circuits by Correlated L1 Sampling
Bounding the cost of classically simulating the outcomes of universal quantum circuits to additive error δ is often called weak simulation and is a direct way to determine when they confer a quantum advantage. Weak simulation of the T+Clifford gateset is BQP-complete and is expected to scale exponentially with the number t of T gates. We constructively tighten the upper bound on the worst-case L1 norm sampling cost to next order in t from O(ξtδ−2) if δ2≫ξ−t to O((ξt−t)δ−2) if δ2≫(ξt−t)−1, where ξt=2∼0.228t is the stabilizer extent of the t-tensored T gate magic state. We accomplish this by replacing independent L1 sampling in the popular SPARSIFY algorithm used in many weak simulators with correlated L1 sampling. As an aside, this result demonstrates that the T gate magic state's approximate stabilizer state decomposition is not multiplicative with respect to t, for finite values, despite the multiplicativity of its stabilizer extent. This is the first weak simulation algorithm that has lowered this bound's dependence on finite t in the worst-case to our knowledge and establishes how to obtain further such reductions in t.
We prove that every pure state of stabilizer rank at most k has stabilizer extent at most 2O(klog(k+1)), and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-k operators. This implies stabilizer fidelity at least 2−O(klog(k+1)), resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an Ω(n2/logn) lower bound for the approximate stabilizer rank of tensor powers of any non-stabilizer qubit state. The latter result generalizes the best-known lower bound for tensor powers of T-states (Mehraban-Tamasbi, STOC, 2024) to arbitrary non-stabilizer qubit states, including magic states.
As a consequence of the Clifford rank--norm inequality, we obtain an Ω(n2/logn) lower bound for exact representations of n-bit AND function by quadratic phases, improving the previous best-known linear bound. Further consequences rule out pseudorandom state and unitary ensembles with approximate stabilizer and Clifford rank O((logn)2/loglogn), respectively, a logn improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most k, with poly(n)2O(klog(k+1)) time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.