# Explicit lower bounds on strong simulation of quantum circuits in terms of $T$-gate count

arXiv: Quantum Physics, 2019.

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Abstract:

We investigate Clifford+$T$ quantum circuits with a small number of $T$-gates. Using the sparsification lemma, we identify time complexity lower bounds in terms of $T$-gate count below which a strong simulator would improve on the state-of-the-art $3$-SAT solving.

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Introduction

- Assuming the ETH, there exists constant a > 0 such that any classical algorithm solving 3-SAT instances with length L takes 2aL time, where again L is the length of the formula.
- Assume that a classical algorithm solves 3-SAT in time O(23.1432×10−7L), where L is the length of the formula, m2 is the number of 2-clauses, m3 is the number of 3-clauses, so that m = m2 + m3 and L = 2m2 + 3m3 − 1.
- The necessity of the Sparsification Lemma comes from reducing the 3-SAT problem to quantum circuits, as the number of T gates in

Highlights

- Assuming the Exponential Time Hypothesis (ETH), there is an > 0 such that any strong simulation that can determine if 0|C|0| = 0 of a polynomial-sized quantum circuit C formed from the Clifford+T gate set with N T -gates takes time at least 2 N
- We review the Exponential Time Hypothesis but for explicit constants, we have the following theorem
- The reduction from 3-SAT to strong simulation of quantum circuits is described in Section 3, while a full proof of the Sparsification Lemma with explicit constants will be given in the Appendix
- (i) For a 3-SAT instance φ with length L, we construct a quantum circuit Cφ with at most 2L Toffoli gates and poly(L) NOT and CNOT gates such that Cφ|x I |0 A|0 B = |x I |0 A|φ(x) B, where A is the system of ancilla qubits and B is a system of one single qubit
- For any > 0 we have a quantum circuit simulator that runs in time O(2 14 N ), where N is the number of T gates and the size of the circuit is O(N )

Results

- The reduction from 3-SAT to strong simulation of quantum circuits is described in Section 3, while a full proof of the Sparsification Lemma with explicit constants will be given in the Appendix.
- (i) For a 3-SAT instance φ with length L, the authors construct a quantum circuit Cφ with at most 2L Toffoli gates and poly(L) NOT and CNOT gates such that Cφ|x I |0 A|0 B = |x I |0 A|φ(x) B, where A is the system of ancilla qubits and B is a system of one single qubit.
- In time polynomial in L, the authors can construct a reversible circuit C that tidily computes φ with at most 2L Toffoli gates.
- Given a 3-SAT formula φ with n variables and length L, one can efficiently construct a quantum circuit Cφ consisting only of Clifford gates and at most 14L T -gates, so that
- According to Corollary 9 a 3-SAT instance of length L reduces to the strong simulation problem of a linear size quantum circuit with Clifford gates and at most N = 14L T gates.
- If there were a solution to the simulation problem with running time O(22.2451×10−8N ), any 3-SAT instance of length L could be solved in time O(22.2451×10−8×14L) = O(23.1432×10−7L), contradicting Lemma 4.
- The set of 3-SAT formulae corresponding to leaf nodes is the collection of instances in , which is the list the authors needed to construct.
- The high-level idea of the proof is as follows: the authors show that there are at most a linear number of immigrant clauses ever introduced.

Conclusion

- Suppose that for each a > 0, contradictory to Lemma 3, there is an algorithm Solvea that solves 3-SAT in time O(2aL).
- Assume the authors had a 3-SAT solver Solve that runs in time o(23.1432×10−7L) on instances of length L.
- Run the Sparsification Algorithm on input φ to get a list of 2γ(θ1,θ2)n sparse instances in time O(2γ(θ1,θ2)npoly(n)), each with length η(θ1, θ2)n.

Summary

- Assuming the ETH, there exists constant a > 0 such that any classical algorithm solving 3-SAT instances with length L takes 2aL time, where again L is the length of the formula.
- Assume that a classical algorithm solves 3-SAT in time O(23.1432×10−7L), where L is the length of the formula, m2 is the number of 2-clauses, m3 is the number of 3-clauses, so that m = m2 + m3 and L = 2m2 + 3m3 − 1.
- The necessity of the Sparsification Lemma comes from reducing the 3-SAT problem to quantum circuits, as the number of T gates in
- The reduction from 3-SAT to strong simulation of quantum circuits is described in Section 3, while a full proof of the Sparsification Lemma with explicit constants will be given in the Appendix.
- (i) For a 3-SAT instance φ with length L, the authors construct a quantum circuit Cφ with at most 2L Toffoli gates and poly(L) NOT and CNOT gates such that Cφ|x I |0 A|0 B = |x I |0 A|φ(x) B, where A is the system of ancilla qubits and B is a system of one single qubit.
- In time polynomial in L, the authors can construct a reversible circuit C that tidily computes φ with at most 2L Toffoli gates.
- Given a 3-SAT formula φ with n variables and length L, one can efficiently construct a quantum circuit Cφ consisting only of Clifford gates and at most 14L T -gates, so that
- According to Corollary 9 a 3-SAT instance of length L reduces to the strong simulation problem of a linear size quantum circuit with Clifford gates and at most N = 14L T gates.
- If there were a solution to the simulation problem with running time O(22.2451×10−8N ), any 3-SAT instance of length L could be solved in time O(22.2451×10−8×14L) = O(23.1432×10−7L), contradicting Lemma 4.
- The set of 3-SAT formulae corresponding to leaf nodes is the collection of instances in , which is the list the authors needed to construct.
- The high-level idea of the proof is as follows: the authors show that there are at most a linear number of immigrant clauses ever introduced.
- Suppose that for each a > 0, contradictory to Lemma 3, there is an algorithm Solvea that solves 3-SAT in time O(2aL).
- Assume the authors had a 3-SAT solver Solve that runs in time o(23.1432×10−7L) on instances of length L.
- Run the Sparsification Algorithm on input φ to get a list of 2γ(θ1,θ2)n sparse instances in time O(2γ(θ1,θ2)npoly(n)), each with length η(θ1, θ2)n.

Reference

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