River Lane Research Heilbronn Quantum Algorithms Meeting 2018 , Cambridge UK riverlane.io

A Generalised Variational Quantum EigensolverDaochen Wang, Oscar Higgott, Stephen Brierley

River Lane ResearcharXiv:1802.00171

I. Introduction

5

• • •

|+iU

= |+i • H⇧

H • H⇧

H

| i | i / P † R† R P R† R

FIG. 4. Circuit used for casting expectation estimation as phase estimation. The eigenphases ±�0 of S0 on |+ i can be used to determine theexpectation of P via cos(±�0/2) =

��� 1+h |P | i2��� .

log n)/✏↵).

C. Generalised ↵-VQE

Finally we generalise VQE by using the result of §II Bto replace each expectation estimation sub-routine in thestandard VQE of Figure 1 by the ↵-QPE developed in§IIA. A schematic of our generalised VQE is given inFigure 5. Henceforth, we refer to generalised VQE run-ning ↵-QPE as ↵-VQE.

Clearly, ↵-VQE still preserves the following three keyadvantages of standard VQE because we only generalisedthe expectation estimation subroutine. First, we can par-allelise the expectation estimation sub-routines to multi-ple quantum computers thereby reducing running time.Second, robustness via self correction is preserved be-cause ↵-VQE is still variational [6, 7]. Third, the vari-ational parameter � can be classically stored to enablestraightforward re-preparation of | (�)i [32]. In the next§III, we discuss ↵-VQE in comparison to VQE and QPE.

FIG. 5. Schematic of generalised VQE. Note that the � parameteralso a↵ects the implementation of ↵-QPE circuits through the presence

of R(�) and R(�)†in the S

0operators involved (see Figure 4). When

↵ = 0, we are in the statistical sampling, or standard VQE, regime.

When ↵ = 1, we are in the phase estimation regime.

III. RESOURCE COMPARISONS

We reiterate that the advantage of ↵-VQE is its abil-ity to perform expectation estimation in regimes laying

continuously between statistical sampling and phase esti-mation. Neither edge regime is typically ideal: statisticalsampling requires N = O(1/✏2) samples whereas phaseestimation requires D = O(1/✏) coherence time. Each ofthese two regimes has been criticised by the other in thisway by Ref. [11] and Ref. [3, 6] respectively, and com-pared in this way in Ref. [32]. ↵-VQE can address suchcriticisms/comparisons by choosing ↵ to trade o↵ N andD according to given costs on each, as we exemplifiedtowards the end of §II A.The resources required for one run of expectation es-

timation within VQE and ↵-VQE (arbitrary ↵, ↵ = 0,↵ = 1) are compared in Table I. The comparison to QPEis less straightforward. At a general level, QPE requiresgreater depth at each repetition but few repetitions to beconfident that the ground state energy has already beenmeasured, even if the prepared state has only mild over-lap with the ground state. This latter advantage owesto the state collapse feature of QPE unavailable to VQEor ↵-VQE (hence the need for a classical optimiser). Weonly remark that the simple argument “QPE scales bet-ter in ✏ against VQE” becomes weaker against ↵-VQE,particularly because ↵-VQE may at least run under lim-ited depth when QPE may not even run. Note that thegreater depth which QPE needs to run is not only due tothe scaling with ✏.Concretely, consider the electronic Hamiltonian H,

written in second quantised form as:

H =X

pq

hpqa†paq +

1

2

X

pqrs

hpqrsa†pa†qaras, (10)

where the indices run over n introduced spin basis or-bitals. With second order Trotter decomposition, imple-menting c-exp(−iHt) for fixed t requires, at first count[33], a circuit depth of O(n11) as follows: O(n4) fromthe number of sub-terms in the second quantised form ofH (10), O(n) from the Jordan-Wigner transformation ofthese sub-terms necessary to preserve Fermionic commu-tation relations, and O(n6) from the Trotter decomposi-tion [17]. Rapid progress has been made in reducing theO(n11) depth scaling. However, the best known depthÕ(n8/3) [8] using plane wave basis functions is still worsethan the best known depth for variational methods ofO(CR) ⌘ O(n) (which ↵-VQE retains because the over-head, due to c-S0, of O(n) adds onto CR).

III. Replace expectation estimation by α-QPE

II. Define α-QPE 3|+i Z(✓) • E 2 {0, 1}

|�i / U2j

FIG. 2. Circuit for Kitaev’s QPE. Here, |�i is an eigenstate of Uwith eigenphase � (i.e. U |�i = ei� |�i), |+i is the +1 eigenstate ofX, Z(✓) := diag(1, e

�i✓), and measurement is performed in the X

basis. We refer to C1 and C2 as this circuit with ✓ = 0 and ✓ =

�⇡/2 respectively. To infer m bits of � with a constant probability ofsuccess, C1 and C2 are each ran O(log(m)) times for j = m � 1, ...., 0in that order. The total number of measurements is therefore N =

O(m log(m)) = Õ(m) and circuit depth is D = O(2mlog(m)) = Õ(2

m)

[16]. Writing m ⌘ log2(1/✏), where ✏ is the error in �, gives N =Õ(log(1/✏)) and D = Õ(1/✏).

naive multiplication [19]. This follows from the additiveperiodic structure of the multiplicative group (Z/FZ)⇤.

Under the framework of Kitaev’s QPE, Wiebe andGranade [20] recently introduced a Bayesian QPE namedRFPE which we now modify to yield di↵erent sets of cir-cuit and measurement sequences that can provide thesame precision ✏ with di↵erent (N,D) tradeo↵s. Thesesets shall be parametrised by the ↵ 2 [0, 1]. The circuitfor RFPE, much like that for Kitaev’s QPE, is given inFigure 3 below and the following presentation of RFPEand our modification is broadly self-contained. For de-tails, see the references cited.

To begin, a prior probability distribution P (�) of � istaken to be normal N (µ,�2) (some justification is givenin Ref. [21] which empirically found that the posterior ofa uniform prior converges rapidly to normal). From theRFPE circuit in Figure 3, we deduce the probability ofmeasuring E 2 {0, 1} is:

P (E|�;M, ✓) = 1 + (�1)Ecos(M(�� ✓))

2, (3)

which enters the posterior by the Bayesian update rule:

P (�|E;M, ✓) / P (E|�;M, ✓)P (�). (4)

We do not need to know the constant of proportion-ality to sample from this posterior after measuring E,and the word “rejection” in RFPE refers to the rejectionsampling method used. After obtaining a number s ofsamples, we approximate the posterior again by a nor-mal with mean and standard deviation equal to that ofour samples (again justified as when taking initial priorto be normal). The choice of s is important and s canbe regarded as a particle filter number, hence the word“filter” in RFPE. We constrain our posteriors to be nor-mal essentially because it allows for e�cient sampling inthe next iteration. For more details on s and rejectionfiltering, see Ref. [20].

The e↵ectiveness of RFPE’s iterative update proce-dure just described depends on controllable parameters(M, ✓). A natural measure of e↵ectiveness is the expectedposterior variance, i.e. the “Bayes risk”. To minimise theBayes risk, Ref. [20] chooses M = d1.25/�e at the start

|+i Z(M✓) • E 2 {0, 1}

| i / UM

FIG. 3. Circuit for Rejection Filtering Phase Estimation (RFPE). Thenotation follows that of Kitaev’s QPE in Figure 2. However, before each

iteration of this circuit, the choice of (M, ✓) can be classically calculated

so as to minimise the expected posterior variance (i.e. Bayes risk) after

measuring E.

of each iteration. However, the main problem is thatM can quickly become large, making the depth of UM

exceed Dmax. Ref. [20] solves this problem by simplyimposing an upper bound on M and we refer to this ap-proach as RFPE-with-restarts. Here, we propose anotherapproach that chooses:

(M, ✓) = (1

�↵, µ� �), (5)

where ↵ 2 [0, 1] is a free parameter we impose. Moreoverat each iteration, we propose a new preparation of eigen-state |�i, discarding that used at the previous iteration.This requires the atypical ability to readily prepare aneigenstate (see §II B). We name the resulting, modifiedRFPE algorithm ↵-QPE and derive Proposition 1 statedbelow in Appendix A. Unlike in Kitaev’s algorithm, “pre-cision ✏” below is the expected posterior standard devia-tion [22].

Proposition 1

For precision ✏, ↵-QPE requires:

N = f(✏,↵), (6)

D =1

✏↵, (7)

where the number of measurements and maximum co-herent depth are given by N and D respectively andthe function f : R⇥ [0, 1] ! R is the defined as:

f(✏,↵) =

(2

1�↵ (1

✏2(1�↵)� 1) if ↵ 2 [0, 1)

4 log( 1✏) if ↵ = 1

. (8)

We now consider ↵-QPE under the constraint 1 D Dmax for some constant Dmax (i.e. D cost is zero untilsome threshold Dmax when it becomes infinite). This isexperimentally realistic when Dmax equals the transversecoherence time T2 and when it is realistic to approxi-mate the standard e�t/T2 model for T2 coherence by astep-function in t that jumps from full coherence to zerocoherence at t = T2.Suppose we require precision ✏ 2 (0, 1) and wish to

minimise N . Then, because N = f(✏,↵) is an decreasingfunction of ↵, the least N is attained at the maximal

This is the α-QPE circuit which is iterated to find an eigenphase � of � . Before each iteration, the real tuple � is chosen as � where � , � is the current mean and standard deviation on the Bayesian posterior of � .

Notation: � , � is the � � eigenstate, measurement in the � basis.

ψ U(M, θ) (1/σα, μ − σ) μ

σψ

Z(Mθ) = diag(1, e−iMθ) | + ⟩ + 1 XX

With � defined by the right circuit, Knill et al. [2] showed that � is always in a 50:50 superposition of two eigenstates of � with eigenphases � respectively where � . Running α-QPE with this � estimates the expectation value � . This differs from standard expectation estimation which uses statistical sampling.

Notation: � , � .

U|ψ⟩

U ± ϕϕ = 2arccos( |1 + ⟨ψ |P |ψ⟩ | /2)

U⟨ψ |P |ψ⟩

Π = I − 2 |0⟩⟨0 | R : |0...0⟩ ↦ |ψ⟩

Ourgeneralisation

6

Algorithm Maximum coherent depth Non-coherent repetitions Total resources

VQE O(CR) O( 1✏2 ) O(CR1✏2)

0-VQE O(CR + log n) O( 1✏2 ) O((CR + log n)1✏2)

1-VQE O((CR + log n) 1✏ ) O(log1✏ ) O((CR + log n)

1✏ )

↵-VQE O((CR + log n) 1✏↵ ) O(f(✏,↵)) O((CR + log n)1✏↵ f(✏,↵))

TABLE I. Resource comparison of one expectation estimation subroutine within VQE, 0-VQE, 1-VQE, ↵-VQE. ✏ is the precision required forthe expected energy, CR is the state preparation depth cost, n is the number of qubits, and ↵ 2 [0, 1] is the free parameter that determines thecircuit depth of ↵-QPE. Note that 0-VQE would never be advantageous over VQE but is included for completeness.

[1] S. Lloyd, Science 1996, 273, 1073–1078.[2] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, M. Head-

Gordon, Science (New York, N.Y.) 2005, 309, 1704–1707.

[3] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q.Zhou, P. J. Love, A. Aspuru-Guzik, J. L. O’Brien, NatureCommunications 2014, 5, ncomms5213.

[4] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, M. Troyer,Proceedings of the National Academy of Sciences of the

United States of America 2017, 114, 7555–7560.[5] B. M. Ho↵man, D. Lukoyanov, Z.-Y. Yang, D. R. Dean,

L. C. Seefeldt, Chemical Reviews 2014, 114, 4041–4062,PMID: 24467365.

[6] J. R. McClean, J. Romero, R. Babbush, A. Aspuru-Guzik, New Journal of Physics 2016, 18, 23023.

[7] P. J. J. O’Malley, R. Babbush, I. D. Kivlichan,J. Romero, J. R. McClean, R. Barends, J. Kelly,P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen,Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jef-frey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley,C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wen-ner, T. C. White, P. V. Coveney, P. J. Love, H. Neven,A. Aspuru-Guzik, J. M. Martinis, Physical Review X2016, 6, 31007.

[8] R. Babbush, N. Wiebe, J. McClean, J. McClain,H. Neven, G. K. Chan, arXiv preprint arXiv:1706.000232017.

[9] R. Santagati, J. Wang, A. A. Gentile, S. Paesani,N. Wiebe, J. R. McClean, S. R. M. Short, P. J. Shadbolt,D. Bonneau, J. W. Silverstone, D. P. Tew, X. Zhou, J. L.OBrien, M. G. Thompson 2017.

[10] N. Wiebe, C. Granade, Physical Review Letters 2016,117, 10503.

[11] S. Paesani, A. A. Gentile, R. Santagati, J. Wang,N. Wiebe, D. P. Tew, J. L. O’Brien, M. G. Thompson,Physical Review Letters 2017, 118, 100503.

[12] One could alternatively bound the circuit area or totalnumber of quantum gates. We use circuit depth for sim-plicity.

[13] E. Knill, G. Ortiz, R. D. Somma, Physical Review A2007, 75, 12328.

[14] J. Romero, R. Babbush, J. R. McClean, C. Hempel,P. Love, A. Aspuru-Guzik 2017.

[15] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, J. M.Chow, J. M. Gambetta, arXiv preprint arXiv:1704.050182017.

[16] A. Y. Kitaev, A. Shen, M. N. Vyalyi, Classical andQuantum Computation, American Mathematical Society,2002.

[17] D. W. Berry, G. Ahokas, R. Cleve, B. C. Sanders, Com-munications in Mathematical Physics 2007, 270, 359–371.

[18] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, R. D.Somma, Phys. Rev. Lett. 2015, 114, 090502.

[19] M. A. Nielsen, I. L. Chuang, Quantum computationand quantum information, Cambridge University Press,2010.

[20] N. Wiebe, C. Granade, A. Kapoor, K. M. Svore, Approx-imate Bayesian Inference via Rejection Filtering, 2015.

[21] C. Ferrie, C. E. Granade, D. G. Cory, Quantum Infor-mation Processing 2013, 12, 611–623.

[22] In Appendix A, we numerically check that this approxi-mates the actual error. An actual standard deviation of✏ on an unbiased posterior mean implies “precision ✏” inKitaev’s sense by Markov’s inequality. The converse isnot true [34].

[23] D. Golovin, A. Krause, D. Ray, Near-optimal Bayesianactive learning with noisy observations, 2010. https://dl.acm.org/citation.cfm?id=2997275.

[24] D. Hsu, PhD thesis UC San Diego 2010.[25] J. O. Berger, Statistical Decision Theory and Bayesian

Analysis, Springer New York, 1985.[26] This is analogous to the assumption that spectral gaps

are at least � > 0 in [20] which facilitates a restartingstrategy.

[27] V. V. Shende, I. L. Markov, S. S. Bullock, Phys. Rev. A2004, 69, 062321.

[28] D. Maslov, Phys. Rev. A 2016, 93, 022311.[29] L. Isenhower, M. Sa↵man, K. Mølmer, Quantum Infor-

mation Processing 2011, 10, 755.[30] K. Mølmer, L. Isenhower, M. Sa↵man, Journal of Physics

B: Atomic, Molecular and Optical Physics 2011, 44,184016.

[31] X.-Q. Zhou, T. C. Ralph, P. Kalasuwan, M. Zhang,A. Peruzzo, B. P. Lanyon, J. L. O’Brien, Nature Com-munications 2011, 2, 413.

[32] D. Wecker, M. B. Hastings, M. Troyer, Physical ReviewA 2015, 92, 42303.

[33] D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings,M. Troyer, Physical Review A 2014, 90, 22305.

[34] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wise-man, G. J. Pryde, Nature 2007, 450, 393–396.

6

Algorithm Maximum coherent depth Non-coherent repetitions Total resources

VQE O(CR) O( 1✏2 ) O(CR1✏2)

0-VQE O(CR + log n) O( 1✏2 ) O((CR + log n)1✏2)

1-VQE O((CR + log n) 1✏ ) O(log1✏ ) O((CR + log n)

1✏ )

↵-VQE O((CR + log n) 1✏↵ ) O(f(✏,↵)) O((CR + log n)1✏↵ f(✏,↵))

TABLE I. Resource comparison of one expectation estimation subroutine within VQE, 0-VQE, 1-VQE, ↵-VQE. ✏ is the precision required forthe expected energy, CR is the state preparation depth cost, n is the number of qubits, and ↵ 2 [0, 1] is the free parameter that determines thecircuit depth of ↵-QPE. Note that 0-VQE would never be advantageous over VQE but is included for completeness.

[1] S. Lloyd, Science 1996, 273, 1073–1078.[2] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, M. Head-

Gordon, Science (New York, N.Y.) 2005, 309, 1704–1707.

[3] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q.Zhou, P. J. Love, A. Aspuru-Guzik, J. L. O’Brien, NatureCommunications 2014, 5, ncomms5213.

[4] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, M. Troyer,Proceedings of the National Academy of Sciences of the

United States of America 2017, 114, 7555–7560.[5] B. M. Ho↵man, D. Lukoyanov, Z.-Y. Yang, D. R. Dean,

L. C. Seefeldt, Chemical Reviews 2014, 114, 4041–4062,PMID: 24467365.

[6] J. R. McClean, J. Romero, R. Babbush, A. Aspuru-Guzik, New Journal of Physics 2016, 18, 23023.

[7] P. J. J. O’Malley, R. Babbush, I. D. Kivlichan,J. Romero, J. R. McClean, R. Barends, J. Kelly,P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen,Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jef-frey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley,C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wen-ner, T. C. White, P. V. Coveney, P. J. Love, H. Neven,A. Aspuru-Guzik, J. M. Martinis, Physical Review X2016, 6, 31007.

[8] R. Babbush, N. Wiebe, J. McClean, J. McClain,H. Neven, G. K. Chan, arXiv preprint arXiv:1706.000232017.

[9] R. Santagati, J. Wang, A. A. Gentile, S. Paesani,N. Wiebe, J. R. McClean, S. R. M. Short, P. J. Shadbolt,D. Bonneau, J. W. Silverstone, D. P. Tew, X. Zhou, J. L.OBrien, M. G. Thompson 2017.

[10] N. Wiebe, C. Granade, Physical Review Letters 2016,117, 10503.

[11] S. Paesani, A. A. Gentile, R. Santagati, J. Wang,N. Wiebe, D. P. Tew, J. L. O’Brien, M. G. Thompson,Physical Review Letters 2017, 118, 100503.

[12] One could alternatively bound the circuit area or totalnumber of quantum gates. We use circuit depth for sim-plicity.

[13] E. Knill, G. Ortiz, R. D. Somma, Physical Review A2007, 75, 12328.

[14] J. Romero, R. Babbush, J. R. McClean, C. Hempel,P. Love, A. Aspuru-Guzik 2017.

[15] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, J. M.Chow, J. M. Gambetta, arXiv preprint arXiv:1704.050182017.

[16] A. Y. Kitaev, A. Shen, M. N. Vyalyi, Classical andQuantum Computation, American Mathematical Society,2002.

[17] D. W. Berry, G. Ahokas, R. Cleve, B. C. Sanders, Com-munications in Mathematical Physics 2007, 270, 359–371.

[18] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, R. D.Somma, Phys. Rev. Lett. 2015, 114, 090502.

[19] M. A. Nielsen, I. L. Chuang, Quantum computationand quantum information, Cambridge University Press,2010.

[20] N. Wiebe, C. Granade, A. Kapoor, K. M. Svore, Approx-imate Bayesian Inference via Rejection Filtering, 2015.

[21] C. Ferrie, C. E. Granade, D. G. Cory, Quantum Infor-mation Processing 2013, 12, 611–623.

[22] In Appendix A, we numerically check that this approxi-mates the actual error. An actual standard deviation of✏ on an unbiased posterior mean implies “precision ✏” inKitaev’s sense by Markov’s inequality. The converse isnot true [34].

[23] D. Golovin, A. Krause, D. Ray, Near-optimal Bayesianactive learning with noisy observations, 2010. https://dl.acm.org/citation.cfm?id=2997275.

[24] D. Hsu, PhD thesis UC San Diego 2010.[25] J. O. Berger, Statistical Decision Theory and Bayesian

Analysis, Springer New York, 1985.[26] This is analogous to the assumption that spectral gaps

are at least � > 0 in [20] which facilitates a restartingstrategy.

[27] V. V. Shende, I. L. Markov, S. S. Bullock, Phys. Rev. A2004, 69, 062321.

[28] D. Maslov, Phys. Rev. A 2016, 93, 022311.[29] L. Isenhower, M. Sa↵man, K. Mølmer, Quantum Infor-

mation Processing 2011, 10, 755.[30] K. Mølmer, L. Isenhower, M. Sa↵man, Journal of Physics

B: Atomic, Molecular and Optical Physics 2011, 44,184016.

[31] X.-Q. Zhou, T. C. Ralph, P. Kalasuwan, M. Zhang,A. Peruzzo, B. P. Lanyon, J. L. O’Brien, Nature Com-munications 2011, 2, 413.

[32] D. Wecker, M. B. Hastings, M. Troyer, Physical ReviewA 2015, 92, 42303.

[33] D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings,M. Troyer, Physical Review A 2014, 90, 22305.

[34] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wise-man, G. J. Pryde, Nature 2007, 450, 393–396.

Resourcecomparisons

Boxed in red: up to square root speed-up

The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm typically used to approximate the ground energy of a Hamiltonian � where � are tensored Pauli operators. It is often compared with the quantum phase estimation algorithm (QPE). Idea: combine them!

Our work replaces the expectation estimation subroutine of VQE by a version of Bayesian QPE [1], which we name α-QPE, in order to reduce the subroutine’s run-time by up to a square root. This is possible by exploiting quantum coherence time.

H = ∑ aiPi Pi

References[1] N. Wiebe, C. Granade, Physical Review Letters 2016, 117, 10503.[2] E. Knill, G. Ortiz, R. D. Somma, Physical Review A 2007, 75, 12328.