Quantum-circuit design for efficient simulations of many-body quantum dynamics
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| Number of Parts | 15 | |
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| License | CC Attribution - NonCommercial - ShareAlike 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this | |
| Identifiers | 10.5446/39075 (DOI) | |
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Transcript: English(auto-generated)
00:07
Our main result is an efficient algorithm to design efficient quantum circuits to simulate quantum many-body Hamiltonians. This algorithm is optimal in number of qubits as well as in the scaling of number of gates.
00:22
Another advantage that we believe is essential for quantum simulation is that the user can specify the error of simulation. We also introduce a method to reduce the circuit depths. This is essential for parallel quantum computation. At the end we present numerical evidences which indicate that current upper
00:41
bounds for quantum simulation overestimate the simulation error by several orders of magnitude. The goal in Hamiltonian simulation algorithms is to emulate the action of dynamics generated by a Hamiltonian on an initial state. Such simulations are important because they are the only known methods for efficiently simulating broad classes of physically relevant quantum systems.
01:06
The first step in these algorithms is to find a mapping between the states in the original Hilbert space and a subspace of the simulator's Hilbert space. This mapping allows a quantum state to be prepared in the simulator that is logically equivalent to the initial state of the simulation. The simulator then performs a sequence of quantum operations to this input state, transforming it to a
01:25
state that is logically equivalent to one that is close to the evolved state of the simulated system. This simulated state is the output of the quantum algorithm. Although the resultant state can generally be learned efficiently, other quantities such as expectation values of local observables can be found efficiently by measuring the state yielded by the algorithm.
01:46
We take the simulator to be a quantum computer that can perform the Hadamard gate, the pi by 8 gate, and the controlled NOT gate. Our objective is to construct a classical algorithm that builds simulation circuits that use as few of these gates as possible.
02:02
The inputs to our algorithm are a bit of string representing the Hamiltonian, the error tolerance for the simulation, and the evolution time. And then in the outputs, a bit of string representing the final circuit that simulates the evolution under the input Hamiltonian up to the error tolerance.
02:20
The input Hamiltonian is promised to be a K-local Hamiltonian, and for the quantum simulator we use a quantum computer that can do a universal set of gates. This process is performed in three steps. First, the Hamiltonian is sorted into a sum of mutually commuting groups of operators to reduce the circuit depth.
02:41
Next, the Lee-Trouder Suzuki algorithm is applied to decompose the evolution into a sequence of exponentials of the Pauli operators. The final step involves designing circuits to implement each of the exponentials. The circuit implementation of them involves switching to the eigenbasis of the exponential, implementing the evolution in the eigenbasis, and then transforming back to the computational basis.
03:03
The resulting circuits are efficient because Pauli operators can be efficiently diagonalized. The idea behind reducing the circuit depths is to sort operations such that those that commute can be done at the same time. Our algorithm points all the commuting operations and groups them together.
03:22
As an example, consider the following circuit. Operation 1 and 4 commute, so our algorithm sorts the operations and groups 1 and 4 together, which reduces the circuit depths. In conclusion, our work provides an efficient algorithm for designing quantum simulation circuits for many-body Hamiltonians.
03:44
We introduce grouping techniques that optimize the depth of the resulting simulation circuits. We also provide numerical estimates of the typical errors that occur from using Trotter Suzuki formulas, and show that existing estimates can be far too loose.
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