Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity

Abstract

We construct quantum circuits which exactly encode the spectra of correlated electron models up to errors from rotation synthesis. By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity O(λ/ ) where λ is an absolute sum of Hamiltonian coefficients and is target precision. For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T gate complexity O(N + log(1/ )) where N is number of orbitals in the basis. This enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity O(N 3 / + N 2 log(1/ )/ ). Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime. Compiling to surface code faulttolerant gates and assuming per gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about a million superconducting qubits in a matter of hours.

Time-dependent Hamiltonian simulation withL1-norm scaling

“…The apparent classical intractability of simulating quantum dynamics led Feynman [25] and others to propose the idea of quantum computation. Quantum computers can simulate various physical systems, including condensed matter physics [3], quantum field theory [29], and quantum chemistry [2,14,47]. The study of quantum simulation has also led to the discovery of new quantum algorithms, such as algorithms for linear systems [28], differential equations [9], semidefinite optimization [11], formula evaluation [22], quantum walk [15], and ground-state and thermal-state preparation [20,42].…”

Improved Quantum Circuits for Elliptic Curve Discrete Logarithms

“…A quantum look-up to N elements requires 4N T-gates [2]. To optimize window costs, we balance this cost against the operations we save.…”

“…For n-bit integers with window size k, repeating this round n/k times performs the full multiplication. Since the quantum look-up will cost 4 · 2 k T gates [2] and uncontrolled n + k-bit addition costs O(n + k) T gates, we expect the optimal window size to be approximately k = O(lg n). The total multiplication cost is still O(n 2 ) because we only window addition by p, not addition of the quantum register y.…”

Magic State Distillation: Not as Costly as You Think

“…If these states are used to perform logical P π/8 rotations, one out of every ∼1/p phys logical gates is expected to be faulty. Since faulty gates spoil the outcome of a computation, but classically intractable quantum computations typically involve more than 10 8 T gates [7], low-error magic states are required to execute gates with a low error probability. One possibility to generate low-error magic states is via a magic state distillation protocol.…”

“…4 9,3,3 × (15-to-1) 25,9,9 10 −4 6.3 × 10 −25 18,630 67. 8 1,260,000 25.9d 3 / d = 29 21.2d 3 / d = 31 (15-to-1) 17,7,7 10 −3 4.5 × 10 −8 4,618 42.6 197,000 6.30d 3 / d = 25 4.04d 3 / d = 29 (15-to-1) 6 13,5,5 × (20-to-4) 21,11,13 10 −3 1.4 × 10 −10 43,344 130 1,410,000 28.9d 3 / d = 29 19.6d 3 / d = 33 (15-to-1) 4 13,5,5 × (20-to-4) 27,13,15 10 −3 2.6 × 10 −11 46,790 157 1,840,000 30.9d 3 / d = 31 21.5d 3 / d = 35 (15-to-1) 6 11,5,5 × (15-to-1) 25,11,11 10 −3 2.7 × 10 −12 30,732 82.5 2,540,000 35.3d 3 / d = 33 25.0d 3 / d = 37 (15-to-1) 6 13,5,5 × (15-to-1) 29,11,13 10 −3 3.3 × 10 −14 39,108 97. 5 3,810,000 37.6d 3 / d = 37 27.7d 3 / d = 41 (15-to-1) 6 17,7,7 × (15-to-1) 41,17, 17 10 −3 4.5 × 10 −20 73,460 128 9,370,000 39.8d 3 / d = 49 31.5d 3 / d = 53 Small-footprint and synthillation protocols (15-to-1) 9,3, 3 10 −4 1.5 × 10 −9 762 36.2 27,600 6.27d 3 / d = 13 4.08d 3 / d = 15 (15-to-1) 9,5,5 × (15-to-1) 21,9,11 10 −3 6.1 × 10 −10 7,782 469 3,650,000 74.7d 3 / d = 29 50.7d 3 / d = 33 (15-to-1) 4 7,3,3 × (8-to-CCZ) 15 Table 1: Comparison of different distillation protocols with respect to the following characteristics: physical error rate p phys , output error probability per output state pout, space cost in qubits, time cost in surface-code cycles, and space-time cost in qubitcycles.…”