Simulation of quantum circuits by low-rank stabilizer decompositions

Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of s t a b i l i z e r rank , which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of χ ∼ 10 6 stabilizer states and sample from the full n -bit output distribution, improving on previous simulations which used ∼ 10 3 stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 T gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of s t a b i l i z e r rank , which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of χ ∼ 10 6 stabilizer states and sample from the full n -bit output distribution, improving on previous simulations which used ∼ 10 3 stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 T gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.

arXiv:1808.00128QuantumSep 2, 2019302 citations

Classical simulations of quantum systems using stabilizer decompositions

Hammam Qassim

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 $O(\\ell n^2)$ to $O(m n^2)$ , 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 $\\ell \\gg m$ . \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 $\\ket{\\psi}^{\\otimes m}$ , where $\\ket{\\psi}$ is either a magic state or an equatorial state. \nFor any single qubit magic state $\\ket{\\psi}$ , we find stabilizer decompositions of $\\ket{\\psi}^{\\otimes m}$ with $2^{m\\log_2(3)/4}$ terms. This improves on the best known bound of $2^{m \\log_2(7)/6}$ . \nSimilarly, for any single qubit equatorial state $\\ket{\\psi}$ , we give a stabilizer decomposition of $\\ket{\\psi}^{\\otimes m}$ with $2^{m/2}$ 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.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 $O(\\ell n^2)$ to $O(m n^2)$, 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 $\\ell \\gg m$. \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 $\\ket{\\psi}^{\\otimes m}$, where $\\ket{\\psi}$ is either a magic state or an equatorial state. \nFor any single qubit magic state $\\ket{\\psi}$, we find stabilizer decompositions of $\\ket{\\psi}^{\\otimes m}$ with $2^{m\\log_2(3)/4}$ terms. This improves on the best known bound of $2^{m \\log_2(7)/6}$. \nSimilarly, for any single qubit equatorial state $\\ket{\\psi}$, we give a stabilizer decomposition of $\\ket{\\psi}^{\\otimes m}$ with $2^{m/2}$ 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.

Feb 19, 20214 citations

Improved upper bounds on the stabilizer rank of magic states

Hammam Qassim, Hakop Pashayan, David Gosset

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.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.

arXiv:2106.07740QuantumDec 20, 202143 citations

Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits

Yifei Huang, Peter J. Love

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 $O({n}^{3})$ algorithm for calculating the inner product of two $n$ -qudit stabilizer states.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 $O({n}^{3})$ algorithm for calculating the inner product of two $n$-qudit stabilizer states.

arXiv:1808.02406Physical Review AMay 6, 201915 citations

Efficient Simulation of High-Level Quantum Gates

Adam Husted Kjelstrøm, Andreas Pavlogiannis, Jaco van de Pol

Quantum circuit simulation is paramount to the verification and optimization of quantum algorithms, and considerable research efforts have been made towards efficient simulators. While circuits often contain high-level gates such as oracles and multi-controlled X ( C k X ) gates, existing simulation methods require compilation to a low-level gate-set before simulation. This, however, increases circuit size and incurs a considerable (typically exponential) overhead, even when the number of high-level gates is small. Here we present a gadget-based simulator which simulates high-level gates directly, thereby allowing to avoid or reduce the blowup of compilation. Our simulator uses a stabilizer decomposition of the magic state of non-stabilizer gates, with improvements in the rank of the magic state directly improving performance. We then proceed to establish a small stabilizer rank for a range of high-level gates that are common in various quantum algorithms. Using these bounds in our simulator, we improve both the theoretical complexity of simulating circuits containing such gates, and the practical running time compared to standard simulators found in IBM's Qiskit Aer library. We also derive exponential lower-bounds for the stabilizer rank of some gates under common complexity-theoretic hypotheses. In certain cases, our lower-bounds are asymptotically tight on the exponent.Quantum circuit simulation is paramount to the verification and optimization of quantum algorithms, and considerable research efforts have been made towards efficient simulators. While circuits often contain high-level gates such as oracles and multi-controlled X ( C k X ) gates, existing simulation methods require compilation to a low-level gate-set before simulation. This, however, increases circuit size and incurs a considerable (typically exponential) overhead, even when the number of high-level gates is small. Here we present a gadget-based simulator which simulates high-level gates directly, thereby allowing to avoid or reduce the blowup of compilation. Our simulator uses a stabilizer decomposition of the magic state of non-stabilizer gates, with improvements in the rank of the magic state directly improving performance. We then proceed to establish a small stabilizer rank for a range of high-level gates that are common in various quantum algorithms. Using these bounds in our simulator, we improve both the theoretical complexity of simulating circuits containing such gates, and the practical running time compared to standard simulators found in IBM's Qiskit Aer library. We also derive exponential lower-bounds for the stabilizer rank of some gates under common complexity-theoretic hypotheses. In certain cases, our lower-bounds are asymptotically tight on the exponent.

arXiv:2507.04337QuantumMay 5, 20260 citations

Improvements of the classical simulation of quantum circuits: Using graph states with local Cliffords

Matthijs S. C. Rijlaarsdam

With this thesis project, we improve the classical simulation of quantum computers using stabilizers in the GSLC formalism. We do this in two ways: we present new algorithms that speed up their simulation and extend their applicability by defining new operations and subroutines for existing general circuit simulation using GSLC. To be precise: we present multiple new algorithms that speed up the simulation of the CZ gates, the most computationally expensive quantum operation in GSLC formalism. We define two new operations on GSLC that are useful when simulating stabilizer circuits: calculating fidelity (a measure of 'closeness' between two quantum states), and tracing out qubits (throwing away the information about the state contained in these qubits) from a GSLC. Finally, we present a new GSLC-based subroutine for a state of the art general quantum circuit simulation algorithm by Bravyi et al. that allows for the usage of the faster CZ algorithms. We show that the GSLC formalism can give a speedup in practical simulation tasks by evaluating the complexity of simulating an algorithm with possible applications on near-term quantum hardware: the quantum approximate optimization algorithm.With this thesis project, we improve the classical simulation of quantum computers using stabilizers in the GSLC formalism. We do this in two ways: we present new algorithms that speed up their simulation and extend their applicability by defining new operations and subroutines for existing general circuit simulation using GSLC. To be precise: we present multiple new algorithms that speed up the simulation of the CZ gates, the most computationally expensive quantum operation in GSLC formalism. We define two new operations on GSLC that are useful when simulating stabilizer circuits: calculating fidelity (a measure of 'closeness' between two quantum states), and tracing out qubits (throwing away the information about the state contained in these qubits) from a GSLC. Finally, we present a new GSLC-based subroutine for a state of the art general quantum circuit simulation algorithm by Bravyi et al. that allows for the usage of the faster CZ algorithms. We show that the GSLC formalism can give a speedup in practical simulation tasks by evaluating the complexity of simulating an algorithm with possible applications on near-term quantum hardware: the quantum approximate optimization algorithm.

Jan 1, 20200 citations

O(\\ell n^2)

O(m n^2)

, 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

\\ell \\gg m

. \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

\\ket{\\psi}^{\\otimes m}

, where

\\ket{\\psi}

is either a magic state or an equatorial state. \nFor any single qubit magic state

\\ket{\\psi}

, we find stabilizer decompositions of

\\ket{\\psi}^{\\otimes m}

with

2^{m\\log_2(3)/4}

terms. This improves on the best known bound of

2^{m \\log_2(7)/6}

. \nSimilarly, for any single qubit equatorial state

\\ket{\\psi}

, we give a stabilizer decomposition of

\\ket{\\psi}^{\\otimes m}

with

2^{m/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.

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({n}^{3})

algorithm for calculating the inner product of two

n

-qudit stabilizer states.