Lee et al. present it as "affordable and systematically improvable" — the second half is the role of $k$, and the first is what the pair restriction buys. They compare against the full generalized set (UCCGSD) and the standard one (UCCSD) on classical benchmarks, and report that "$k$-UpCCGSD is found to show the best scaling for quantum computing applications" of the three.

The Hamiltonian whose ground state is wanted, together with whatever structure is to be respected — particle number, spin, point-group symmetry, a reference determinant — and the connectivity and native gate set of the device the family has to run on.

1. Hamiltonian whose eigenvalues are wanted → Parameterised circuit familygeneralize the singles, pair the doubles

   The wavefunction is built as a $k$-fold product of independent unitary exponentials acting on a reference determinant: $|\psi\rangle=\Pi_{\alpha=1}^{k}\left(e^{\hat T^{(\alpha)}-\hat T^{(\alpha)\dagger}}\right)|\phi_0\rangle$ (Eq. 15), where each $\hat T^{(\alpha)}=\hat T_1+\hat T_2$ carries its own independent set of singles and doubles amplitudes. $\hat T_1=\frac{1}{2}\sum_{pq}t_p^q\hat a_q^\dagger\hat a_p$ ranges over all spin-orbitals (the paper's default index convention, stated at the top of Sec. 2), not just occupied-to-virtual. $\hat T_2=\sum_{ia}t_{i_\alpha i_\beta}^{a_\alpha a_\beta}\hat a^\dagger_{a_\alpha}\hat a^\dagger_{a_\beta}\hat a_{i_\beta}\hat a_{i_\alpha}$ assumption: pairs the two spin-orbitals sharing a spatial orbital $i$ and moves both together into a shared spatial orbital $a$, so the amplitude indices run over spatial, not spin, orbitals. approximation: this pair restriction discards the rest of the generalized-doubles manifold that UCCGSD keeps, buying a circuit depth that grows only as $O(kN)$ instead of $O(N^3)$, with $k$ the dial that restores the accuracy the restriction gives up.

   approximationassumption

A circuit family with a fixed structure and free real parameters, together with the number of those parameters — which is the size of the classical search problem handed to the next layer.

given  Hamiltonian H on N spin-orbitals, mapped 1-1 onto N qubits (qubit count is
       linear in N),  eta electrons,  reference determinant |phi_0>
       (this paper's own benchmarks use a restricted-Hartree-Fock determinant),
       and an integer k >= 1 supplied by the caller

requires  each factor's amplitude set is independent of every other factor's --
    #   no amplitude is shared across l = 1 .. k
requires  k is not fixed here by a formula in N, eta or a target error -- the paper
    #   only reports, after the fact, that k must be taken large enough to reach
    #   sub-chemical accuracy, or the energy surface loses the smoothness that makes
    #   it usable for tasks like scanning it ("Summary of Chemical Applications")

# --- k-fold product of independent pair-restricted unitaries -----------------
for l = 1 .. k:
    draw amplitude set  { t_p^q }  over ALL spin-orbital pairs p, q       # generalized singles
    draw amplitude set  { t_(iup,idown)^(aup,adown) }  over occupied spatial orbitals i,
        virtual spatial orbitals a                                       # paired doubles
        # iup/idown and aup/adown are the two spin components of one spatial orbital,
        #   not the loop index l

    T1^l  =  (1/2) sum_pq  t_p^q  adag_q a_p                                    (Eq. 12)
        # p, q run over occupied AND virtual spin-orbitals alike -- 'generalized',
        #   not restricted to occupied-to-virtual as in ordinary UCCSD (Eq. 4)
    T2^l  =  sum_ia  t_(iup,idown)^(aup,adown)  adag_aup adag_adown a_idown a_iup   (Eq. 14)
        # moves both spins of spatial orbital i into spatial orbital a together --
        #   the pCCD-style restriction that drops the rest of the generalized-doubles
        #   manifold UCCGSD keeps
    T^l  =  T1^l + T2^l
    U^l  =  exp( T^l - (T^l)^dag )

return  |psi(theta)>  =  U^1 U^2 ... U^k |phi_0>                                (Eq. 15)
    # Eq. 15 itself is only written  Pi_(l=1)^k exp(T^l - (T^l)^dag) |phi_0>  --
    #   no time-ordering symbol, no stated convention for which factor acts first;
    #   since each U^l carries an independent amplitude set the U^l's do not
    #   commute in general, so operator order is a real degree of freedom the
    #   source leaves open, not a resolved detail. Expanded here the way a Sigma
    #   would be read -- left to right in l -- so U^k acts on |phi_0> first and
    #   U^1 last; this ordering is this listing's reading, not a claim the paper
    #   itself makes
    # theta is the full stack of k independent {singles, paired-doubles} amplitude
    #   sets; k is supplied here, not derived -- it is the caller's dial, not this
    #   construction's output

# --- circuit realization, at a fixed number of Trotter steps per U^l (Table 1) --
gate count  =  O(k N^2)
    # total gate count is lower-bounded by the number of cluster amplitudes, and
    #   each factor's singles + paired-doubles amplitudes number O(N^2)   (Table 1)
group each U^l's paired-doubles terms into O(N) sets of O(N) terms acting on disjoint
    spin-orbitals;  execute every set in parallel on a linearly-connected qubit array
    # this is what reduces the O(kN^2) gate count to a shallower circuit -- it assumes
    #   maximal parallelization of terms on distinct spin-orbitals and neglects
    #   Jordan-Wigner overhead (the derivation's own stated assumption, not a
    #   general-purpose compiler guarantee)

Circuit depth $\mathcal{O}(kN)$ in the number $N$ of spin orbitals and the number $k$ of repetitions, as stated in the abstract, against $\mathcal{O}(N^3)$ for UCCGSD and $\mathcal{O}((N-\eta)^2\eta)$ for UCCSD with $\eta$ electrons. A depth for the family, not a complexity for solving the problem.

• k-UpCCGSD ansatz · Qiskit

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  Repeated paired generalized doubles with generalized singles trade expressivity against shallower chemistry circuits.

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• k-UpCCGSD ansatz

  Repeated paired generalized doubles with generalized singles trade expressivity against shallower chemistry circuits.