Variational Quantum Computation of Excited StatesOscar Higgott1,2, Daochen Wang1,3, and Stephen Brierley1

1 Riverlane, 3 Charles Babbage Road, Cambridge CB3 0GT2 Department of Physics and Astronomy, University College London, London, WC1E 6BT3Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742

July 1, 2019

The calculation of excited state energies ofelectronic structure Hamiltonians has many im-portant applications, such as the calculation ofoptical spectra and reaction rates. While low-depth quantum algorithms, such as the varia-tional quantum eigenvalue solver (VQE), havebeen used to determine ground state ener-gies, methods for calculating excited states cur-rently involve the implementation of high-depthcontrolled-unitaries or a large number of addi-tional samples. Here we show how overlap esti-mation can be used to deflate eigenstates oncethey are found, enabling the calculation of ex-cited state energies and their degeneracies. Wepropose an implementation that requires thesame number of qubits as VQE and at most twicethe circuit depth. Our method is robust to con-trol errors, is compatible with error-mitigationstrategies and can be implemented on near-termquantum computers.

1 IntroductionEigenvalue problems are ubiquitous in almost all fieldsof science and engineering. Google’s PageRank al-gorithm alone has had a significant impact on mod-ern society, and at its core solves an eigenvalue prob-lem associated with a stochastic matrix describing theWorld Wide Web [28]. Another important example isPrincipal Component Analysis (PCA) [13, 29], whichhas widespread applications in bioinformatics, neuro-science, image processing, and quantitative finance.

The time-independent Schrodinger equation providesyet another example of a fundamental eigenvalue prob-lem. Its numerical solution enables properties of atoms,molecules and materials to be predicted, with far-reaching applications in materials design, drug discov-ery and fundamental science [38]. Characterisation ofexcited state energies of molecules is required to predictcharge and energy transfer processes in photovoltaic

Oscar Higgott: oscar.higgott@gmail.com

materials, or to understand some chemical reactions,such as those that involve photodissociation. However,classical methods such as density functional theory areoften unable to determine excited states, even for ma-terials where ground state energy calculations are pos-sible.

Quantum computers have the potential to solve theseand other problems significantly faster than any knownmethods using classical computers [1, 11, 19, 36]. How-ever many quantum algorithms will require quantumerror correction, limiting their usefulness in the nearfuture [31]. Here we study hybrid quantum-classical al-gorithms, which dramatically reduce the required gatedepth to run and somewhat mitigate errors, by closelyintegrating classical and quantum subroutines [2, 9, 14,15, 21, 23, 26, 40].

The variational quantum eigensolver (VQE), intro-duced in Ref. [30], is the first algorithm designed tofind the lowest eigenvalue of a Hamiltonian on a near-term, non-fault-tolerant quantum computer. VQE isbased on the variational principle and utilises the factthat quantum computers can store quantum states us-ing exponentially fewer resources than required classi-cally. VQE uses parameterised quantum circuits to pre-pare trial wavefunctions and compute their energy, anda classical computer to find the parameters minimisingthis energy. The low circuit depth of VQE has led to thehope that it may enable near-term quantum-enhancedcomputation.

Since its introduction, modifications have been sug-gested to enable VQE to find excited state ener-gies: e.g. a folded spectrum method [30] which re-quires finding the expectation of the squared Hamil-tonian with quadratically more terms, or symmetry-based methods which are non-systematic [23]. Suchsuggestions have been more recently superseded bytwo proposals: a method that minimises the von Neu-mann entropy [35] and the quantum subspace expansionmethod [5, 24]. However, the von Neumann entropymethod (“WAVES”) requires a large number of high-depth controlled-unitaries, and the quantum subspaceexpansion method requires a large number of additional

Accepted in Quantum 2019-06-14, click title to verify 1

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samples compared to VQE and introduces a new ap-proximation.

Our algorithm extends VQE to systematically findexcited states at almost no extra cost. We achieve thisby adding “overlap” terms onto the optimisation func-tion in order to exploit the fact that Hermitian ma-trices admit a complete set of orthogonal eigenvectors.Exploiting further the fact that VQE retains the clas-sical parameters of ansatz states that enable their re-preparation, low-depth quantum circuits can then bereadily used to calculate these overlap terms.

2 Variational quantum deflation algo-rithmIn VQE, the real parameters λ for the ansatz state|ψ(λ)〉 are classically optimised with respect to the ex-pectation value:

E(λ) := 〈ψ(λ)|H |ψ(λ)〉 =∑j

cj 〈ψ(λ)|Pj |ψ(λ)〉 , (1)

of the Hamiltonian H =∑cjPj , computed using a low-

depth quantum circuit. As a result of the variationalprinciple, finding the global minimum of E(λ) is equiv-alent to finding the ground state energy of H. VQEhas been implemented on many experimental platforms,and has been shown to be more resilient to control er-rors than the quantum phase estimation algorithm [27].

Our method extends VQE to calculate the k-th ex-cited state by instead optimising the parameters λk forthe ansatz state |ψ(λk)〉 such that the cost function:

F (λk) := 〈ψ(λk)|H |ψ(λk)〉+k−1∑i=0

βi |〈ψ(λk)|ψ(λi)〉|2 ,

(2)is minimised. This can be seen as minimising E(λk)subject to the constraint that |ψ(λk)〉 is orthogonal tothe states |ψ(λ0)〉 , ..., |ψ(λk−1)〉. In the next section, weshow how choosing sufficiently large β0, ..., βk−1 meansthe minimum of F (λk) is guaranteed to be the energyof the k-th state, provided that the ansatz is sufficientlyexpressive.

While the first term in Eq. (2) is E(λk), and canbe computed using the same quantum circuits as usedfor VQE, the second term is a sum of overlaps of theansatz state with states 0 to k − 1, and can be com-puted efficiently on a quantum computer using one ofthe methods given in Section 4.

Note that evaluating Eq. (2) requires knowledge ofλ0, ..., λk−1 and so an iterative procedure is required tocalculate the k-th eigenvalue. First, λ0 is calculatedusing VQE by minimising E in Eq. (1). Then, λ1 is

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QUANTUM CIRCUITS CLASSICAL CIRCUITS

VARIATIONAL QUANTUM DEFLATION ALGORITHM (VQD)

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Figure 1: A schematic of our variational quantum deflationmethod for finding the k-th excited state of a Hamiltonian H.

calculated by minimising F in Eq. (2) for k = 1, afterwhich λ2 can be determined using the same procedurewith the known λ0 and λ1, and so on until λk is deter-mined.

A schematic of our variational quantum deflation(VQD) algorithm is shown in Fig. 1. An initial guess ofλk is used to generate a state preparation circuit R(λk)that prepares the state |ψ(λk)〉 when applied to the fidu-cial state |0〉. This circuit is used repeatedly to computeeach of the expectation values 〈ψ(λk)|Pj |ψ(λk)〉 (see

Refs. [30, 40]) and overlap terms |〈ψ(λk)|ψ(λi)〉|2 fori < k. The overlap terms are computed using circuitsdescribed in Section 4 or Appendix B or by followingthe method in Ref. [17].

A classical computer then uses the results of thesequantum computations to calculate the objective func-tion F (λk) of Eq. (2) and update λk using a classi-cal optimiser. The new λk is then used to prepare anew ansatz state on the quantum computer, and thewhole process is repeated until some stopping criterionis reached.

As shown in Appendix A, the total number of samplesM (k) required to measure the VQD objective functionto precision ε when finding the kth excited state (as-suming states 0 . . . k − 1 can be perfectly prepared) isbounded above by:

M (k) ≤ 1ε2

L−1∑j=0|cj |+

12

k−1∑i=0

βi

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compared to the VQE sampling cost of M ≤1ε2

(∑L−1j=0 |cj |

)2. For well-chosen βi, we expect this ad-

ditional sampling cost relative to VQE to be very small,

Accepted in Quantum 2019-06-14, click title to verify 2

as explained in more detail in Appendix A.

3 Overlap weightingAn equivalent viewpoint of our optimisation procedureis that we are finding the ground state of the effectiveHamiltonian at stage k:

Hk := H +k−1∑i=0

βi |i〉 〈i| , (4)

where |i〉 is the (previously found) i-th eigenstate of Hwith energy Ei := 〈i|H |i〉 1. It can be easily verifiedthat for an arbitrary state |ψ〉 :=

∑ai |i〉:

〈ψ|Hk |ψ〉 =k−1∑i=0|ai|2(Ei + βi) +

d−1∑i=k|ai|2Ei,

where d is the total number of eigenvectors of H.Therefore, if the ansatz is sufficiently powerful, then

to guarantee a minimum at Ek, it suffices to chooseβi > Ek−Ei. Since ∆ := Ed−1−E0 ≥ Ek−Ei, it sufficesto possess an accurate estimate of ∆, e.g. by using VQEto find E0 and then Ed−1 (using the Hamiltonian −H tofind the latter). When we readily have a specification ofH =

∑cjPj as a linear combination of Pauli matrices,

e.g. when H is the electronic structure Hamiltonian,then we have the upper bound ∆ ≤ 2‖H‖ ≤ 2

∑|cj |.

In this case, we can readily choose βi to guarantee thevalidity of our procedure.

Choosing valid βi can also be self-correcting. For ex-ample, if we incorrectly chose βi = γ − Ei ≤ Ek − Eifor all i, we will discover that we have set βi too smallsince we will eventually find a minimum at F (λk) = γ.However, by repeating the algorithm with a larger γuntil an energy strictly less than γ is found (doublingγ each time, say), we can pick a large enough γ afterO(log (Ek − E0)) runs of the algorithm.

4 Low-depth implementationsA low-depth method for overlap estimation, proposedin Ref. [12], can be seen by writing the overlap| 〈ψ(λi)|ψ(λk)〉 |2 as | 〈0|R(λi)†R(λk) |0〉 |2. We can pre-pare the state R(λi)†R(λk) |0〉 using the trial statepreparation circuit followed by the inverse of the prepa-ration circuit for the i-th previously-computed state.The overlap is then estimated to precision ε by thefraction of all-zero bitstrings when measuring this stateO(1/ε2) times in the computational basis.

1We assume these are true eigenstates with possibly non-distinct energies.

This method requires knowing the inverse of thepreparation circuit for each previously-computed state,R(λi)†. While this inverse is often known in theoryby inverting gates in a decomposition of the originalpreparation circuit, device errors may mean that theimplementation is inaccurate in practice. If we defineλ∗i to be the optimal parameters originally found to pre-pare the i-th state R(λ∗i ) |0〉 using VQD, then its inversecan be found by fixing λ∗i and varying the trial state pa-rameters λi such that the overlap | 〈0|R(λi)†R(λ∗i ) |0〉 |2is maximised. This technique enables VQD to retainthe robustness to control errors that is characteristic ofVQE [27].

This implementation of VQD requires the same num-ber of qubits as VQE and around twice the circuitdepth. In Appendix B, we describe an alternativemethod which uses the destructive SWAP test and re-quires almost the same circuit depth as VQE but twicethe number of qubits. If a larger gate-depth is avail-able, then α-QPE [40] can be used to reduce the totalruntime of overlap estimation from O( 1

ε2 ) up to O( 1ε ).

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(Ha)

ExactVQD

Figure 2: All ground and excited state energy levels of H2 in theSTO-3G basis, calculated using exact diagonalisation (blue dot-ted line) and our variational quantum deflation (VQD) method(red filled circles) over a range of internuclear separations.

5 Numerical simulation: H2

We simulated VQD on H2 in the STO-3G basis for arange of internuclear separations and compared it toexact diagonalisation, as shown in Fig. 2. Using βi = 3Ha for all i and a generalised unitary coupled cluster sin-gles and doubles (UCCGSD) ansatz, the median error ofour method relative to exact diagonalisation is less than

Accepted in Quantum 2019-06-14, click title to verify 3

4×10−6 Ha for all energy levels, significantly better thanchemical accuracy of 1.6 × 10−3 Ha (the precision re-quired to determine reaction rates to within an order ofmagnitude at room temperature using the Eyring equa-tion [8]). Our method finds all 6 eigenstates systemat-ically, including all those in the 3-dimensional degener-ate subspace spanned by the 1st, 2nd and 3rd excitedstates. The ability to find degenerate states is anotherkey advantage of our method; the folded spectrum andWAVES methods rely on the energies of states to dif-ferentiate between them and have no systematic way ofdetermining the degeneracy of the eigenvalues. Furtherdiscussion of our simulation, including optimiser andansatz used, can be found in Appendix C.

6 Error accumulationIn general, we cannot assume perfect state preparationfor states i < k. Suppose a state |ψ0〉 (with energyE0) is prepared instead of the true ground state |ψ0〉such that

∣∣〈ψ0|ψ0〉∣∣2 = 1 − ε0, leading to an error in

the ground state energy ε′0 = E0 − E0 = O(2ε0||H||).If we now use this ground state estimate along withVQD to find the first excited state |ψ1〉 using a newtrial state |ψ1〉, the lowest-energy state of the deflatedHamiltonian no longer corresponds to the exact excitedstate energy E1.

The inexact deflated Hamiltonian is now given byH1 = H + β0 |ψ0〉 〈ψ0| and, to assess the accumulationof errors, we wish to find upper and lower bounds forminψ1

[〈ψ1| H1 |ψ1〉].In Appendix D we show that, provided we set β0 >

E1−E01−ε0

, the ground state energy of the inexact deflatedHamiltonian is bounded by terms linear in ε0:

E1−O((E1−E0)ε0) ≤ minψ〈ψ| H1 |ψ〉 ≤ E1 +β0ε0. (5)

In reality we will not find the exact ground state of thedeflated Hamiltonian H1 and will incur an additionalerror ε1 as was the case for the ground state of theoriginal Hamiltonian H. However, provided ε1 ≈ ε0,our total error ε′1 = O(ε0β0 + 2ε1||H||) in the energyis still linear in our original ground state error. Analternative analysis of error accumulation is providedby Lee et al. [18].

We analysed this accumulation of errors furtherthrough numerical simulations of VQD in the presenceof sampling error, shown in Fig. 3. We analysed threedifferent sampling rates: M = 106, 107 and 108 sam-ples per Hamiltonian subterm and overlap term, run-ning 225 simulations of VQD (with random initial pa-rameters) for each of these three scenarios. Of theseruns, ∼ 20% of the simulations found the eigenstates inthe incorrect order and were discarded for consistency

0 1 2 3 4 5State (k)

10−4

10−3

Med

ian error (Ha

)

Chemical accuracy108107106

Figure 3: Median error using VQD to determine each energylevel k in the spectrum of H2 at bond distance (0.7414 A) inthe STO-3G basis. Red, blue and green lines show results using106, 107 and 108 samples (per Hamiltonian subterm and over-lap term) respectively. For the solid lines, the standard VQDalgorithm was used, whereas for dashed lines states i < k inHk were computed exactly for comparison. Chemical accuracy(1.6 × 10−3 Ha) is also shown for reference (black solid line).Error bars show 1σ standard errors for the median estimates(calculated using bootstrap resampling [6]).

in the analysis. The median errors for the remaining∼ 180 runs for each state k are shown in Fig. 3. Forcomparison, we also simulated 130 runs (dashed lines)using ‘exact‘ states i < k in Hk (< 10−7 energy errorin each state i < k). For all three sampling rates, themedian error in the first excited state is similar in mag-nitude to the error in the ground state, as expected fromour analysis earlier in this section and in Appendix D.Furthermore, the median errors for all states k < 4 arevery similar (for a given M) to the errors when using anexact Hk, and are all below chemical accuracy, demon-strating that error accumulation is negligible for thesestates. For k = 4 and k = 5 the accumulated error issubstantially higher than the error using an exact Hk,however, showing that VQD is most effective for low-lying states. Achieving chemical accuracy for k = 5requires 107 samples, instead of 106 for 0 < k < 4.

One way to address this accumulation of errors withinVQD to find higher excited states may be to use the al-ternative effective Hamiltonians discussed in Section 7.Another solution is to use a hybrid approach, usingVQD instead of excitation operators in the WAVES pro-tocol [35]. Here, VQD may provide a more effectivemethod of approximating excited states than the exci-tation operators proposed in WAVES, whereas the von-Neumann entropy “eigenstate witness” used in WAVESdoes not have the same problem of error accumulation,

Accepted in Quantum 2019-06-14, click title to verify 4

and could help refine the energy estimate. Both of thesealternative approaches require a larger gate depth thanthe version of VQD we have studied here, but may bea good approach to finding higher excited states in theera of fault-tolerant quantum computing.

7 Choice of effective Hamiltonian

The form of our effective Hamiltonian in Eq. (4) is onlyone choice within the broad category of deflation meth-ods. Such methods are typically employed to find eigen-values and eigenvectors of positive semi-definite matri-ces, often covariance matrices in the context of PCA,starting from the largest eigenvalues.

To make direct use of deflation methods for posi-tive semi-definite matrices, note that the HamiltonianH ′ := −H + E′ for some E′ ≥ Ed−1, e.g. E′ = ‖H‖,is positive semi-definite. Under this transformation,we find that Hotelling’s deflation corresponds to ourmethod and would set βi = E′ − Ei in Eq. (4).

Other deflation methods exist such as projection de-flation or Schur complement deflation which are de-signed to address the problem of not obtaining trueeigenstates at each stage. These two methods, in con-trast to Hotelling’s, ensure that the true ground state ofthe effective Hamiltonian at each stage does not overlapwith the previously found eigenstate estimate irrespec-tive of its accuracy. Empirically, these two methodshave been found to perform better than Hotelling’s inthe context of PCA on some datasets [20].

For example, in projection deflation, the effectiveHamiltonian at stage k is defined as:

Hk = A†k(H − E′)Ak, (6)

where:

Ak :=k−1∏i=0

(1− |i〉 〈i|) ≈ 1−k−1∑i=0|i〉 〈i| , (7)

and the last approximation holds when the previouslyfound eigenvectors |i〉 are truly orthogonal.

With this approximation, writing H again as a lin-ear combination of Pauli matrices Pj , the value of〈ψ|Hk |ψ〉 for an ansatz |ψ〉 is a linear combination ofterms of forms: 〈ψ|Pj |ψ〉, |〈ψ | i〉|2 as in Hotelling’sdeflation, but now additionally 〈ψ | i〉, 〈ψ|Pj |i〉 and〈i|Pj |l〉 (for i, l < k). Without this approximation, wealso need to calculate 〈i | l〉. The important point nowis that all these additional terms can still be quantumcomputed, e.g. following the method in Ref. [17].

8 DiscussionWe have introduced a new method–variational quan-tum deflation (VQD)–for calculating low-lying excitedstate energies of quantum systems using a quantumcomputer. Our method requires the same number ofqubits as the variational quantum eigensolver (VQE)for ground state methods, at most twice the maximumcircuit depth (for any given ansatz) and a negligible in-crease in the number of required measurements. By con-trast, existing methods for quantum computing excitedstates require a large overhead in resources comparedto ground state methods.

While we used a Nelder-Mead optimiser and UCCSDansatz in our simulation of molecular Hydrogen here,we note that many other optimisers and ansatz circuitscan also be used for VQD. After the first version ofthis paper was released, interesting work by Jones et al.compared the use of two different optimisation methodsas applied to our protocol to calculate the spectrum of aLithium Hydride molecule [15]. More recently, work byLee et al. showed that using a multi-determinental ref-erence state or their k-UpCCGSD ansatz can improvethe precision of finding the first excited state of N2 us-ing VQD [18]. Further work could include numericalanalysis of different optimisers and ansatz circuits foruse within VQD in the presence of noise, as well asthe effectiveness of the alternative effective Hamiltoni-ans presented in Section 7.

Given its low resource requirements and compatibil-ity with error-mitigation techniques, we hope that VQDmay enable the quantum-enhanced computation of ex-cited state energies in the near-future.

A Sampling costIn VQE, the variance ε2 in the energy expectation value〈H〉 after using Mj samples for the measurement ofeach subterm 〈Pj〉 in the Hamiltonian H =

∑cjPj is

bounded by [30, 33]:

ε2 =L−1∑j=0

c2jσ

2j

Mj(8)

=L−1∑j=0

c2j (1− 〈Pj〉

2)Mj

≤L−1∑j=0

c2j

Mj. (9)

where σ2j = Var [〈Pj〉] = 〈P 2

j 〉 − 〈Pj〉2

is the intrinsicvariance of the projective measurement of 〈Pj〉. Usingthe method of Lagrange multipliers, Rubin et al. [33]showed that the optimal choice of Mj to minimise thetotal number of samples M =

∑jMj used to achieve

Accepted in Quantum 2019-06-14, click title to verify 5

precision ε is:

Mj = 1ε2|cj |σj

L−1∑i=0|ci|σi, (10)

which leads to a total number of samples

M = 1ε2

L−1∑j=0|cj |σj

2

≤ 1ε2

L−1∑j=0|cj |

2

. (11)

Assuming perfect state preparation for states i < k inVQD, we find that the variance of the energy expecta-tion value 〈Hk〉 of the deflated Hamiltonian Hk is in-stead given by:

ε2 =L−1∑j=0

c2jσ

2j

Mkj

+k−1∑i=0

β2i σi

2

Mki

(12)

≤L−1∑j=0

c2j

Mkj

+k−1∑i=0

β2i

4Mki

(13)

where Mkj is the number of samples used for measuring

〈Pj〉, Mki is the number of samples used to estimate the

overlap of the ansatz with the ith previously found stateand σi

2 = |〈i|k〉|2 (1 − |〈i|k〉|2) is the intrinsic varianceof this overlap measurement. From a straightforwardextension of the Lagrange multiplier approach used byRubin et al. [33], we now find the optimal Mk

j and Mki

for the deflated Hamiltonian Hk to be:

Mkj = 1

ε2|cj |σj

(L−1∑l=0|cl|σl +

k−1∑i=0

βiσi

), (14)

Mki = 1

ε2βiσi

L−1∑j=0|cj |σj +

k−1∑l=0

βlσl

. (15)

This leads to a total number of samples M (k) given by:

M (k) =L−1∑j=0

Mkj +

k−1∑i=0

Mki (16)

= 1ε2

L−1∑j=0|cj |σj +

k−1∑i=0

βiσi

2

(17)

≤ 1ε2

L−1∑j=0|cj |+

12

k−1∑i=0

βi

2

. (18)

From a comparison of M (k) with M we expect thesampling overhead of VQD relative to VQE to be verysmall, since the sum of the L = O(N4) Hamiltonian co-efficients will likely be far larger than the sum of well-chosen weights βi for low-lying excited states in prac-tice. Furthermore, the variances σi

2 tend to zero at con-vergence. However, if we require precision ε throughout

the optimisation rather than just at convergence, andif we choose βi = 2

∑L−1j=0 |cj | since this always guaran-

tees βi is large enough, then we find M (k) = (1 + k)2M(where we have used the upper bounds for both M (k)

and M).

B Destructive SWAP testThe SWAP test enables the overlap |〈φ|ψ〉|2 of twostates |ψ〉 and |φ〉 to be determined to precision ε us-ing O(1/ε2) repeated measurements after applying acircuit to a quantum register in the state |ψ〉 ⊗ |φ〉.While the original SWAP test acting on two N -qubitstates required an ancilla and a controlled-SWAP gate,leading to a 2N + 1-qubit circuit with depth O(N),it was shown in Refs. [4, 10] that the same outcomedistribution can be attained more efficiently withoutan ancilla, using parallel Bell-basis measurements andclassical logic. This so-called “destructive SWAP test”(shown in Fig. 4) requires just 2N qubits and depthO(1), achieving significant savings compared to theoriginal SWAP test.

|0i

R(~�i)

• H · · · mi0

⇡⇡

|0i • H · · · mi1

⌧⌧

|0i · · · • H min

''mi · mk (mod 2)

|0i

R(~�k)

· · · mk0

77

|0i · · · mk1

??

|0i · · · mkn

FF

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Figure 4: The N -qubit generalisation of the destructive SWAPtest as applied to two ansatz states |ψ(λi)〉 and |ψ(λk)〉, pre-pared using state preparation circuits R(λi) and R(λk) respec-tively.

If the ansatz used can be implemented on a linearchain of qubits with nearest neighbour connectivity,e.g. parameterised adiabatic state preparation using thefermionic SWAP network Trotter step [16], then theSWAP test to compare two ansatz states can be im-plemented on a N × 2 nearest-neighbour grid quantumcomputer architecture with a depth-one circuit that issubgraph isomorphic to the architecture (i.e. no routingof quantum information required). This implementa-tion makes the assumption that the same ansatz statecan be prepared with the same parameters on two sepa-rate registers of qubits. If this cannot be assumed (e.g. ifqubit errors are inhomogeneous), then the SWAP testcan be used to “copy” the state from the first registerto the second register, by maximising the overlap of the

Accepted in Quantum 2019-06-14, click title to verify 6

two states, with the parameters of the state on the sec-ond register allowed to vary. This technique allows theSWAP test implementation of VQD to maintain robust-ness to control errors.

C Methods for numerical simulationThe standard UCCSD ansatz [32] is defined relative toa reference state |ψ0〉 by:

|ψ〉 = eT−T†|ψ0〉 ,

where T := T1 + T2 with:

T1 :=∑i∈occl∈vir

tlia†mai,

T2 :=∑

i,j∈occl,k∈vir

tlkija†l a†kaiaj ,

for some parameters tli, tlkij ∈ R and occ and vir are the

sets of occupied and virtual orbitals of |ψ0〉.We instead use a generalised unitary coupled cluster

ansatz (UCCGSD) with |ψ0〉 = |HF〉 set to the Hartree-Fock state but with the cluster operator T = T1 + T2now using the definitions:

T1 :=∑pq

tqpa†qap,

T2 :=∑pqrs

trspqa†ra†sapaq,

where p, q, r, s can now index any orbital (irrespectiveof its occupation in the reference state). A variant ofthis ansatz was suggested by McClean et al. [23] in thecontext of VQE, and UCCGSD has since been investi-gated numerically [18, 41]. Lee et al. found UCCGSDto perform significantly better than UCCSD in VQE fora number of small molecules [18].

Since we are only interested in the parameterisa-tion of T − T †, and fermionic operators obey the anti-commutation relations:

{aj , ak} = 0, {a†j , a†k} = 0, {aj , a†k} = δkj ,

it can be directly verified that there are only 6 and 3 in-dependent parameters for T1 (e.g. t10, t

20, t

30, t

21, t

31, t

32)

and T2 (e.g. t2301, t

1302, t

1203) respectively.

The results in Fig. 2 were simulated using ProjectQand FermiLib [25, 37]. A tolerance of 10−2 was usedwith a Nelder-Mead optimiser (xatol=fatol=10−2, asimplemented in the scipy Python scientific library), andthe best of two consecutive (randomly initialised) runswas used for each bond length and energy level. We note

that other optimisers, such as LGO, have been shown tooffer improved performance in VQE [23], and possiblefurther work includes analysis of alternative optimisa-tion strategies in the context VQD.

While we initialised the UCCGSD parameters ran-domly in this work, choosing a good initial guess for theparameters can significantly reduce the number of itera-tions required for the optimiser to converge. For groundstate VQE problems, second order Møller-Plesset per-turbation theory (MP2) has previously been proposedas a UCC ansatz initialisation method [32]. Anothermethod, for either ground or excited states, initialisesa UCC ansatz with optimised expectation values thatare classically estimated using a truncated BCH expan-sion [18].

We also note that the overlap terms | 〈ψ(λk)|ψ(λi)〉 |2in Eq. (2) of the state k with a known state i are sim-ilar to the overlap terms | 〈ψ(λts)|ψ(λi)〉 |2 of the sameknown state i with another previously-computed states (where i < s < k) in the t-th iteration of the VQDoptimisation procedure used to compute that state. Itmay therefore be advantageous to cache the outputs ofthese | 〈ψ(λts)|ψ(λi)〉 |2 terms, and use them to informand improve the optimisation procedure for the k-thstate, hopefully reducing the number of optimisationsteps and quantum circuits required.

D Bounds for error accumulationIn Section 6 we stated that, to assess the accumulationof errors, we would like to find upper and lower boundsfor the ground state energy minψ[〈ψ| H1 |ψ〉] of the inex-act deflated Hamiltonian H1 = H + β0 |ψ0〉 〈ψ0|, whereψ0 is the (inexact) estimate of the ground state foundin the first iteration of VQD.

Using the same notation as in Section 6, and writ-ing states in the eigenbasis of the Hamiltonian, |ψ0〉 =∑d−1i=0 ai |ψi〉 and |ψ1〉 =

∑d−1i=0 bi |ψi〉, an O(β0ε0) up-

per bound is given straightforwardly by 〈ψ1| H1 |ψ1〉 ≤E1 + β0ε0. Writing a→1 := (a1, . . . , ad−1) and b→1 :=(b1, . . . , bd−1) for compactness, we find the lower boundto be:

〈ψ1| H1 |ψ1〉 (19)

= |b0|2E0 +d−1∑i=1|bi|2Ei + β0|a∗0b0 + 〈a→1 , b→1 〉|2 (20)

= |b0|2(E0 + β0|a0|2) +d−1∑i=1|bi|2Ei

+ β0(2<(a∗0b0〈a→1 , b→1 〉∗) + |〈a→1 , b→1 〉|2) (21)≥ |b0|2(E0 + β0(1− ε0))− |b0|(2β0

√ε0)

Accepted in Quantum 2019-06-14, click title to verify 7

+ minb→

1

d−1∑i=1|bi|2Ei (22)

≥ |b0|2(β0(1− ε0)− (E1 − E0))− |b0|(2β0

√ε0) + E1 (23)

≥ E1 − ε0β2

0β0(1− ε0)− (E1 − E0) . (24)

where the first inequality is Cauchy-Schwarz, the sec-ond inequality follows from

∑d−1i=0 |bi|2 = 1, and the

third inequality follows by minimising a quadratic over|b0| (assuming β0 > E1−E0

1−ε0). From the Taylor series

expansion in ε0 of the second inequality we find:

〈ψ1| H1 |ψ1〉 ≥ E1 −β0(E1 − E0)β0 − (E1 − E0)ε0 +O(ε20), (25)

from which it is clear that, for any fixed β0 >E1−E0

1−ε0,

we have a lower bound of:

minψ1

〈ψ1| H1 |ψ1〉 ≥ E1 −O((E1 − E0)ε0). (26)

E Symmetry constraintsIt is often the case that the Hilbert space of the Hamil-tonian being considered is larger than the Hilbert spacerelevant to the particular problem of interest. For ex-ample, consider the electronic structure Hamiltonian insecond quantised form:

H =∑ij

hija†iaj +

∑ijkl

hijkla†ia†jakal, (27)

where a†i and ai are the fermionic creation and anni-hilation operators for an electron in the i-th spin or-bital, and where the coefficients hij and hijkl denote theone- and two-electron integrals, respectively. After theHamiltonian is transformed through the Jordan-Wigneror Bravyi-Kitaev transformation, converting creationand annihilation operators into qubit operators, the di-mension of the Hilbert space remains 2N , where N is thenumber of spin orbitals. However, if one is interestedonly in states with a particular symmetry, the dimen-sion of the Hilbert space restricted only to these statescan be much smaller, e.g.

(Nη

)= O(Nη) instead of 2N

if only η-electron states are of interest.

If we wish to apply VQD to find excited states of amolecular Hamiltonian with a particular symmetry, itis necessary that the ansatz state for a desired excitedstate, at the global minimum of Eq. (2), be containedentirely within the restricted Hilbert space of interest.One way of ensuring this is to use an ansatz that al-ways conserves the correct symmetry. For example, the

fermionic unitary coupled cluster ansatz we use in Sec-tion 5 conserves the desired number of electrons (η = 2)of neutral molecular Hydrogen for all input parameters.

Alternatively, penalty terms can be included in theobjective function such that the ansatz state has thedesired symmetry at the global minimum of the objec-tive function [23, 34]. This leads to a modified objectivefunction:

FC(λk) := F (λk) +∑i

µi[〈ψ(λk)| Ci |ψ(λk)〉 − ci

]2,

(28)where Ci are symmetry constraining operators (e.g. Ne,

S2, Sz) and ci are constants corresponding to their de-sired expectation values.

Clearly, by incorporating any of these techniques, wecan find the excited states of a Hamiltonian constrainedto any particular symmetry of interest.

F Error mitigationIn Refs. [3, 22, 34], an error-mitigating post-processingprocedure was introduced that uses the operators Ci(defined in Appendix E) to detect and discard all mea-surements that violate a required symmetry for energyexpectation circuits in VQE-type algorithms. This pro-cedure can produce more accurate expectation values inthe presence of bit-flip errors and some combinations oftwo-qubit errors.

After the first version of this paper was released,Ref. [15] incorporated our VQD technique to calcu-late excited states using imaginary time evolution. Theauthors also proposed a method to detect symmetry-breaking errors when using the ancilla-based SWAP-test, by performing symmetry measurements on theansatz registers while measuring the overlap with theancilla. However, using the low-depth overlap estima-tion circuit given in Section 4, we can detect and dis-card any error that does not commute with a symmetryoperator Ci using classical post-processing alone, pro-vided that Ci is diagonal in the computational basis andcommutes with the ansatz. In the Jordan-Wigner andBravyi-Kitaev encodings, the operators for the num-ber of electrons Ne, spin up electrons N↑ and spin

down electrons N↓ are diagonal in the computationalbasis, allowing these quantities to be computed classi-cally in post-processing for both encodings. For exam-ple, starting from |0〉, the UCC ansatz is prepared byRUCC(λ) = V (λ)RHF, where RHF prepares the Hartree-

Fock state |HF〉 and V := eT−T†

is the UCC opera-tor. Now, rather than measuring the fraction of all-zerobitstrings after performing RUCC(λi)†RUCC(λk) |0〉 =R†HFV (λi)†V (λk)RHF |0〉, we can instead measure the

Accepted in Quantum 2019-06-14, click title to verify 8

fraction of bitstrings corresponding to |HF〉 after per-forming V (λi)†V (λk)RHF |0〉. Since V conserves elec-tron number, we know that all measured bitstrings thatdo not correspond to the correct electron number can bediscarded as per the post-processing procedure. There-fore, this method for error-mitigated overlap estimationis more efficient than the ancilla-based method proposedin Ref. [15] if Ci is diagonal in the computational ba-sis. We also note that the error-mitigation techniquesproposed in Refs. [7, 39] can be readily applied to ouralgorithm.

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