Quantum phase estimation

Hamiltonian simulation is rarely used as a standalone algorithm. For the purposes of quan- tum chemistry, its main application is as a subroutine in the quantum phase estimation algorithm (QPEA)231,263_{, which is sometimes referred to as the “von Neumann trick”}273_.

It gives an exponential advantage to many of the algorithms reviewed in the next section. Given a unitary operator eiΦ _{and an approximate eigenstate of that operator prepared on the}

quantum computer, the quantum phase estimation algorithm is used to efficiently output a binary representation of the corresponding phase Φ. In our case, the operator corresponds to a unitary evolution eiHt _{and phase estimation is used to extract information about the}

spectrum of the Hamiltonian.

Figure 3: Circuit performing quantum phase estimation with three ancilla qubits. First, the Hadamard gate (labeled “Had”) is applied on each ancilla qubit to create a uniform superposition. Next, a series of controlled operations U that corresponds to Hamiltonian evolution e−iHt _{are applied. Finally, the inverse QFT is used, see 4.1.3. The input is given}

by one register (consisting of the top three qubits in this specific diagram) initialized to zero and a second register containing the desired eigenstate of H. (The general case where the second register contains a superposition of eigenstates is discussed in the main text.) After execution, the first register stores an approximation of the eigenenergy of _|ψmi. Please see

AppendixC.4 or Nielsen and Chuang1 _{for an introduction to quantum circuits.}

A circuit representation of the algorithm is shown in Figure 3. The QPEA uses two separate registers of qubits.

The first register of T ancilla qubits is used to read out the binary representation of λm. For simplicity, assume that the state |ψmi in the second register is an eigenstate of the

unitary U for which we want to compute the eigenenergy λm.

After each ancilla qubit is initialized in the state |0i, a Hadamard gate is applied to each to prepare this register in an equal superposition of all computational basis states _√1

2T

P

x|xi.

In the first step of reading out λm, the phase (e−iλmt)2

k

is imprinted on the kth ancilla qubit as

(_{|0i + |1i) |ψ}mi →



|0i + e−2kiλmt

This is achieved via “phase-kickback”34,263 _{by applying a controlled-U}2k

between this qubit and the state preparation register in _|ψmi. Finally, the inverse quantum Fourier transform

(see Section 4.1.3) is applied to the ancilla register to convert the state to a computational basis state _|x1x2. . . xTi that expresses the binary representation of λmt/2π ≈ 0.x1x2. . . xT.

After execution of the algorithm, measurement of the first register will yield a binary ap- proximation of the eigenenergy λm with a high probability. It can be shown that with

probability at least 1 − , the estimation of λm is accurate to T − dlog 2 + _21
e
bits1.

Roughly speaking, each ancilla adds another digit of accuracy.

Let us now examine the case when the second register is not an eigenstate but rather a superposition P

mam|ψi. One can perform the above analysis and convince oneself that

phase estimation will yield P

mam

eλmt/2π E

|ψmi where eλm is a fixed-point approximation

of the eigenenergy λm. After measuring of the first register, the second register will collapse

in the corresponding eigenstate. The probability of obtaining the eigenstate λm is |am|2, i.e.

the squared overlap between the eigenstate with the prepared superposition. Therefore, it is possible to obtain multiple eigenstates and eigenenergies of H by running QPEA repeat- edly with an appropriately chosen initial state _{|φi, highlighting the importance of the state} preparation step, as discussed in Section 4.2.1.

QPEA is closely linked to Hamiltonian simulation algorithms because of the controlled- U2k _{transformations. Each of these transformations can be decomposed into a sequence}

of gates by first decomposing U2k _{into a sequence of gates using Hamiltonian simulation}

methods from Section 4.1, then augmenting each gate into a controlled gate with a certain ancilla qubit. Accordingly, advances in quantum algorithms for Hamiltonian simulation lead to improvements in the performance of QPEA. We note, however, that certain methods, such as quantum random access memory (quantum RAM)274_{, could provide an alternative}

means of implementing the powers of controlled unitaries.

In the standard QPEA, the accuracy of the estimated eigenvalue is determined by the number of ancilla qubits. As detailed by Aspuru-Guzik et al.45 _{and Dobˇs´ıˇcek et al.}275_{, the}

number of ancilla qubits can be greatly reduced, while maintaining the same precision by using the iterative quantum phase estimation algorithm (IPEA). Feeding back on the rotation angle of the quantum gates, the phase estimation is improved in each step.

Further improvements to the quantum phase estimation algorithm have been made by introducing Bayesian inference techniques276,277_{. In this way, the maximum amount of infor-}

mation is extracted from previous measurements in order to inform the rotation angles for the next evaluation of the algorithm. Wiebe and Granade277 _{have shown that the eigenenergy}

and its uncertainty can thus be inferred directly rather than iteratively. This Bayesian esti- mation approach to quantum phase estimation has recently been shown experimentally278_,

demonstrating its robustness to noise and decoherence.

The use of the time evolution operator in phase estimation is not necessary for the quantum phase estimation algorithm. Recent work279–281 _{has investigated using unitaries}

which encode the spectrum of the Hamiltonian, but are easier to implement than e−iHt_.

As long as the alternative unitary is determined by a known invertible function f of the Hamiltonian, as e−if (H)t_{, then the measured eigenphase of this unitary can be used to infer}

the corresponding eigenvalue of H. The unitary ei arccos(H/λ) _{can be implemented using a}

quantum walk operator189 _{which requires fewer gates than time evolution.}