The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called “magic” state like |T⟩⊗ n, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and n number of T gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the “exact” rank of |T⟩⊗ n, meaning the minimal size of a decomposition that exactly produces the state. For the “approximate” rank, which is more realistically related to the cost of simulating quantum circuits, no lower bound better than Ω(√n) has been known. In this paper, we improve the lower bound on the approximate rank to Ω(n2) for a wide range of the approximation parameters. An immediate corollary of our result is the existence of polynomial time computable functions which require a super-linear number of terms in any decomposition into exponentials of quadratic forms over F2, resolving a question by Williams. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure, a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure, and a result about trading Clifford operations with T gates.
The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called “magic” state like |T⟩⊗ n, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and n number of T gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the “exact” rank of |T⟩⊗ n, meaning the minimal size of a decomposition that exactly produces the state. For the “approximate” rank, which is more realistically related to the cost of simulating quantum circuits, no lower bound better than Ω(√n) has been known. In this paper, we improve the lower bound on the approximate rank to Ω(n2) for a wide range of the approximation parameters. An immediate corollary of our result is the existence of polynomial time computable functions which require a super-linear number of terms in any decomposition into exponentials of quadratic forms over F2, resolving a question by Williams. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure, a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure, and a result about trading Clifford operations with T gates.
The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩=∑j=1rcj|φj⟩ for cj∈C and stabilizer states φj . The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n -th tensor power of single-qubit magic states.We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(n) of Bravyi, Smith and Smolin \cite{BSS16}. Further, we prove that for a sufficiently small constant δ , the stabilizer rank of any state which is δ -close to those states is Ω(n/logn) . This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F2n , and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.
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.
Recent algorithms for classical simulation of quantum mechanics have runtime whose superpolynomial component is given by a linear dependence on the number of terms required to write large tensor products of certain "magic" states as superpositions of "free" states (which may be stabilizer states, fermionic Gaussian states or others). Surprisingly little is known about the number of terms in such decompositions, called ranks (e.g. the stabilizer rank, fermionic-Gaussian rank etc.). For complexity theoretic reasons they are expected to grow exponentially in the number of tensor factors but, while exponential upper bounds are known for both the stabilizer and fermionic-Gaussian rank of magic states; the best known lower bounds on the stabiliser rank are quadratic, and no bounds on the fermionic-Gaussian rank are known beyond fixed constants. In this work we prove that any fermionic Gaussian decomposition of ∣M⟩⊗k consists of Ω(1.4k) terms. Here ∣M⟩ is the most standard magic state for fermionic linear optics: the 4 qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the δ-approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of 1−δ2. Our results on exact fermionic Gaussian rank apply directly to any product of k fixed parity non-Gaussian states, although the same is not true of the approximate rank. Finally, we prove that the coherent state border rank of an n-mode bosonic Fock state with mj bosons in mode j is exactly ∏j(1+mj), answering a conjecture of Ref. [1] and obtain lower bounds on the approximate coherent state rank given by the same quantity multiplied by a function of the fidelity of the approximation, emphasizing the broad applicability of the method we employ.
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.
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.
The approximate coherent state rank is the minimal number of (classical) coherent states required to approximate a continuous-variable bosonic quantum state and directly relates to the classical complexity of simulating bosonic computations. Despite its importance, little is known about lower bounds on this quantity, even for basic families of states. In this work, we initiate a systematic study of lower bounds on the approximate coherent state rank. Our contributions are as follows.
(i) We introduce a technique based on low-rank approximation theory yielding generic lower bounds on the approximate coherent state rank of arbitrary single-mode states.
(ii) Using this technique, we find a complete characterization of all single-mode states of finite approximate coherent state rank, and we obtain in particular analytical expressions for the approximate coherent state rank of squeezed states and of finite superpositions of Fock states.
(iii) We further show that our single-mode lower bounds can be lifted to multimode lower bounds for finite superpositions of multimode Fock states.
(iv) Finally, we prove a super-polynomial lower bound on the approximate coherent state rank of the n-mode Fock state ∣1⟩⊗n, by exploiting a connection to the permanent. To do so, we show that the algebraic complexity of approximate multi-linear formulas for the permanent is super-polynomial, building upon the proof of a lower bound for exact formulas due to [Raz, JACM 2009].
Our results establish an unconditional barrier to efficient classical simulation of Boson Sampling via coherent state decompositions and connect non-classicality of bosonic quantum systems to central questions in algebraic complexity.