Abstract
We propose examples of a hybrid quantum-classical simulation where a classical computer assisted by a small quantum processor can efficiently simulate a larger quantum system. First we consider sparse quantum circuits such that each qubit participates in O(1) two-qubit gates. It is shown that any sparse circuit on n+k qubits can be simulated by sparse circuits on n qubits and a classical processing that takes time $2^{O(k)} poly(n)$. Secondly, we study Pauli-based computation (PBC) where allowed operations are non-destructive eigenvalue measurements of n-qubit Pauli operators. The computation begins by initializing each qubit in the so-called magic state. This model is known to be equivalent to the universal quantum computer. We show that any PBC on n+k qubits can be simulated by PBCs on n qubits and a classical processing that takes time $2^{O(k)} poly(n)$. Finally, we propose a purely classical algorithm that can simulate a PBC on n qubits in a time $2^{c n} poly(n)$ where $c\approx 0.94$. This improves upon the brute-force simulation method which takes time $2^n poly(n)$. Our algorithm exploits the fact that n-fold tensor products of magic states admit a low-rank decomposition into n-qubit stabilizer states.
Highlights
Quantum computers promise a substantial speed-up over classical ones for certain number-theoretic problems and the simulation of quantum systems [1,2,3]
In the special case when all two-qubit gates in the d-sparse quantum computation on n þ k qubits are CNOTs, we show that the number of repeated n-qubit computations in Theorem 1 scales as 22kðdþ1Þ
We prove that Pauli-based computation (PBC) can be simulated on a classical computer alone more efficiently than one could expect naively
Summary
Quantum computers promise a substantial speed-up over classical ones for certain number-theoretic problems and the simulation of quantum systems [1,2,3]. The PBC model naturally appears in fault-tolerant quantum computing schemes based on error correcting codes of stabilizer type [10] Such codes enable a simple fault-tolerant implementation of nondestructive Pauli measurements on encoded qubits, for example, using the Steane method [11]. Theorem 2.—Any quantum computation in the circuitbased model with n qubits and polyðnÞ gates drawn from the Clifford þ T set can be simulated by a PBC on m qubits, where m is the number of T gates, and polyðnÞ classical processing. Each T gate of the circuit is converted into a simple gadget that includes adaptive Pauli measurements and consumes one copy of the jHi state Simulating such generalized PBCs by the standard PBC on m qubits proves. A lower bound χn ≥ Ωðn1=2Þ is proved in Appendix C
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