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Classification of Stabilizer Simulators
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| # [Classification of Stabilizer Simulators](@id classification_of_stabilizer_simulators) | ||
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| Brute-force simulation algorithms, such as *Schrödinger-style* ([fatima2021faster](@cite)), *Feynman-style* | ||
| ([de2019massively](@cite), [markov2008simulating](@cite), [de2007massively](@cite)), and *hybrid* simulators | ||
| ([markov2018quantum](@cite)) provide high precision for simulating *universal* quantum circuits, but | ||
| they can become highly resource-intensive for circuits with moderate width (around *40* qubits) or depth. | ||
| Alternatively, efficiently classically simulable quantum circuits, like stabilizer circuits, can be | ||
| simulated using the *Gottesman-Knill* theorem, allowing the simulation of thousands of qubits with hundreds | ||
| of thousands of gates. Research to overcome the limitations of these methods falls into two main categories: | ||
| *Born rule probability estimators* that use a quasi-probabilistic representation of the density matrix, and | ||
| *pure-state sampling simulators*. | ||
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| ### Aaronson and Gottesman's Simulator | ||
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| - Introduced in *Improved Simulation of Stabilizer Circuits* [aaronson2004improved](@cite) | ||
| - **Efficient for Stabilizer Circuits**: Introduced a classical simulation algorithm efficient | ||
| for stabilizer circuits. | ||
| - **Simulates Non-Stabilizer Circuits**: Handles non-stabilizer circuits with **exponential** | ||
| run-time cost depending on the number of non-stabilizer gates. | ||
| - **Limitation**: The run-time does not depend on the specific properties of the additional | ||
| non-stabilizer gates, incurring a heavy penalty even for small deviations from stabilizer gates. | ||
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| ### Research Categories to Overcome Limitation | ||
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| #### **Born Rule Probability Estimators** | ||
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| - **Quasi-Probabilistic Representation** | ||
| - *Quantifying quantum speedups: improved classical simulation from tighter magic monotones* ([seddon2021quantifying](@cite)) | ||
| - *From estimation of quantum probabilities to simulation of quantum circuits* ([pashayan2020estimation](@cite)) | ||
| - *On the classical simulability of quantum circuits* ([pashayan2019classical](@cite)) | ||
| - *Estimating outcome probabilities of quantum circuits using quasiprobabilities* ([pashayan2015estimating](@cite)) | ||
| - *Simulation of Qubit Quantum Circuits via Pauli Propagation* ([rall2019simulation](@cite)) | ||
| - *Application of a resource theory for magic states to fault-tolerant quantum computing* ([howard2017application](@cite)) | ||
| - *Negative Quasi-Probability as a Resource for Quantum Computation* ([veitch2012negative](@cite)) | ||
| - *Positive Wigner functions render classical simulation of quantum computation efficient* ([mari2012positive](@cite)) | ||
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| #### **Pure-State Sampling Simulators** | ||
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| - **Bravyi and Gosset Algorithms** | ||
| - *Improved classical simulation of quantum circuits dominated by Clifford gates* ([bravyi2016improved](@cite)) | ||
| - *Efficient Inner-product Algorithm for Stabilizer States* ([garcia2012efficient](@cite)) | ||
| - *On the geometry of stabilizer states* ([garcia2017geometry](@cite)) | ||
| - *Trading classical and quantum computational resource* ([bravyi2016trading](@cite)) | ||
| - *Simulation of quantum circuits by low-rank stabilizer decompositions* ([bravyi2019simulation](@cite)) | ||
| - *Fast Estimation of Outcome Probabilities for Quantum Circuits* ([pashayan2022fast](@cite)) | ||
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| ### Quasi-Probabilistic Simulators | ||
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| - **Purpose**: Produce additive precision estimates of Born rule probabilities. | ||
| - **Representation**: Density matrices are expressed as a linear combination of a preferred set of operators (frame). | ||
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| #### **Frame Choices**: | ||
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| - **Examples**: | ||
| - **Weyl-Heisenberg displacement operators** | ||
| - *From estimation of quantum probabilities to simulation of quantum circuits* ([pashayan2020estimation](@cite)) | ||
| - *On the classical simulability of quantum circuits* ([pashayan2019classical](@cite)) | ||
| - *Simulation of Qubit Quantum Circuits via Pauli Propagation* ([rall2019simulation](@cite)) | ||
| - **Stabilizer states** | ||
| - *Quantifying quantum speedups: improved classical simulation from tighter magic monotones* ([seddon2021quantifying](@cite)) | ||
| - *Application of a resource theory for magic states to fault-tolerant quantum computing* ([howard2017application](@cite)) | ||
| - **Phase-point operators** | ||
| - *Estimating outcome probabilities of quantum circuits using quasiprobabilities* ([pashayan2015estimating](@cite)) | ||
| - *Negative Quasi-Probability as a Resource for Quantum Computation* ([veitch2012negative](@cite) | ||
| - *Positive Wigner functions render classical simulation of quantum computation efficient* ([mari2012positive](@cite)) | ||
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| #### **Special Mention**: | ||
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| - **Dyadic Frame Simulator**: | ||
| - Introduced in *Quantifying quantum speedups: improved classical simulation from tighter magic monotones* ([seddon2021quantifying](@cite)) | ||
| - Decomposes density matrices into stabilizer *dyads* ``|L\rangle\langle R|``. | ||
| - Circuits promoted to universality using magic states. | ||
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| #### **Run-Time**: | ||
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| - Depends quadratically on the dyadic negativity. | ||
| - Dyadic negativity measures deviation from convex combinations of stabilizer dyads. | ||
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| ### Bravyi and Gosset Algorithms | ||
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| - **BG-Estimation Algorithm**: Produces multiplicative precision estimates of Born rule probabilities. | ||
| - **BG-Sampling Algorithm**: Samples approximately from the quantum circuit outcome distribution. | ||
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| #### **Methodology**: | ||
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| - **State Representation**: Initial states are expressed as a linear combination of stabilizer states. | ||
| - **Efficiently Simulable Circuits**: | ||
| - Superposition of polynomially many stabilizer states. | ||
| - Clifford gates and computational basis measurements. | ||
| - **Promoting Universality**: Allowing magic states in initial conditions. | ||
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| #### **Run-Time Dependence**: | ||
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| - Linear in the stabilizer rank of the state. | ||
| - **Stabilizer Rank**: Minimal number of stabilizer states required to represent the state as a linear combination. | ||
| - **Approximation and Stabilizer Extent**: Approximate stabilizer rank can be bounded by the *stabilizer extent* | ||
| ``\xi`` divided by ``\epsilon^2``, where ``\epsilon`` is the approximation error. | ||
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| ### Sum Over Cliffords Algorithm | ||
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| - Introduced in *Simulation of quantum circuits by low-rank stabilizer decompositions* ([bravyi2019simulation](@cite)) | ||
| - **Variant**: Simulates non-Clifford gates using a linear combination of Clifford gates. | ||
| - **Run-Time**: Scales linearly with the stabilizer extent of states like ``|T_\phi^\dagger \rangle``. | ||
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| ### Mixed-State Stabilizer Rank Simulator | ||
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| - Introduced in *Quantifying quantum speedups: improved classical simulation from tighter magic monotones* ([seddon2021quantifying](@cite)) | ||
| - **Improvements**: | ||
| - Generalized BG-sampling algorithm to include mixed states. | ||
| - Improved run-time dependence on error tolerance for approximate sampling. | ||
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| #### **Mixed-State Extent**: | ||
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| - **Definition**: Quantity governing run-time for mixed-state simulators. | ||
| - **Comparison**: For $n$-qubit product states, dyadic negativity, stabilizer extent, and mixed-state extent are equivalent. |
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