



01875nas a2200217   4500008004100000245008100041210006900122260001400191300001100205490000600216520122600222653002701448653003101475100002401506700001301530700002301543700001301566700001801579700002301597856003701620       2022                        eng d00aOptimal scaling quantum linear systems solver via discrete adiabatic theorem0 aOptimal scaling quantum linear systems solver via discrete adiab  c10/7/2022  a0403030 v33 a<p>Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian. Such approaches enabled near-linear scaling in the condition number κ of the linear system, without requiring a complicated variable-time amplitude amplification procedure. However, the most efficient of those procedures is still asymptotically sub-optimal by a factor of log(κ). Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete time evolutions. We use this discrete adiabatic theorem to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in κ, matching a known lower bound on the complexity. Our O(κlog(1/ϵ)) complexity is also optimal in terms of the combined scaling in κ and the precision ϵ. Compared to existing suboptimal methods, our algorithm is simpler and easier to implement. Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.</p>10aFOS: Physical sciences10aQuantum Physics (quant-ph)1 aCosta, Pedro, C. S.1 aAn, Dong1 aSanders, Yuval, R.1 aSu, Yuan1 aBabbush, Ryan1 aBerry, Dominic, W.  uhttps://arxiv.org/abs/2111.0815201759nas a2200169   4500008004100000245008100041210006900122260001500191520123200206100002401438700001301462700002301475700001301498700001801511700002301529856003701552       2021                        eng d00aOptimal scaling quantum linear systems solver via discrete adiabatic theorem0 aOptimal scaling quantum linear systems solver via discrete adiab  c11/15/20213 a<p>Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian. Such approaches enabled near-linear scaling in the condition number κ of the linear system, without requiring a complicated variable-time amplitude amplification procedure. However, the most efficient of those procedures is still asymptotically sub-optimal by a factor of log(κ). Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete time evolutions. We use this discrete adiabatic theorem to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in κ, matching a known lower bound on the complexity. Our O(κlog(1/ε)) complexity is also optimal in terms of the combined scaling in κ and the precision ε. Compared to existing suboptimal methods, our algorithm is simpler and easier to implement. Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.&nbsp;</p>1 aCosta, Pedro, C. S.1 aAn, Dong1 aSanders, Yuval, R.1 aSu, Yuan1 aBabbush, Ryan1 aBerry, Dominic, W.  uhttps://arxiv.org/abs/2111.08152