Gard et al. put the argument for the family before the family itself: "the efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states ... that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them", and "symmetries play a central role in determining the best trial states". Their circuits "respect particle number, total spin, spin projection, and time-reversal symmetries" and "contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space" — minimality OVER THE SUBSPACE, which is a stronger claim than a small parameter count. The construction is general rather than tabulated: they "show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers" and "provide explicit decompositions and gate counts in terms of standard gate sets in each case". The evidence is simulation and its scope is stated: they "test our circuits in quantum simulations of the $H_2$ and $LiH$ molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth".

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 familycascade A gates across the symmetry subspace

   Fixing the particle number maps the target state onto the subspace $H_{n,m}=\mathrm{span}\{|s_1,\dots,s_n\rangle : s_i\in\{0,1\},\sum_i s_i=m\}$, of dimension $\binom{n}{m}$. $X$ gates first place the all-zero reference into this subspace; a cascade of nearest-neighbor two-qubit gates $A(\theta,\phi)$, which acts as the identity on $|00\rangle,|11\rangle$ and rotates $|01\rangle\leftrightarrow|10\rangle$ by $\cos\theta,\ e^{\pm i\phi}\sin\theta$, then generates every superposition within $H_{n,m}$ using the minimal $2\binom{n}{m}-2$ real parameters. assumption: the specific layered A-gate arrangement that reaches this minimal, exactly-spanning circuit for general $n,m$ has no analytical proof -- the paper confirms it only through extensive numerical fidelity checks and states it as a conjecture for arbitrarily many orbitals and electrons. Setting $\phi=0$ in every $A$ gate restricts the state to real coefficients; reaching time-reversal symmetry's minimal parameter count, half of $2\binom{n}{m}-2$, additionally requires fixing one $\theta$ parameter to zero.

   assumption

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  n spin-orbitals from a Jordan-Wigner-mapped Hamiltonian, m electrons fixed
       reference register |0>^(x)n

       A(theta,phi):  two-qubit gate, identity on |00>,|11>; on {|01>,|10>}:
           [ cos theta                exp(i phi) sin theta  ]
           [ exp(-i phi) sin theta       -cos theta          ]        (Eq. 2)
       = 2 single-qubit rotations + 3 CNOTs, minimal in CNOT count    (Fig. 2)

requires  0 < m < n,  linear nearest-neighbor qubit connectivity
    # m=0 or m=n: H_{n,m} is one-dimensional, no ansatz needed

# --- place the reference in the m-excitation sector --------------------------
k = m                          # excitations actually placed by the cascade
if m > n/2:
    k = n - m                  # work in the hole picture, not the particle one
        # gate counts for m and n-m are identical by particle-hole symmetry, so
        #   this is w.l.o.g.; it is REQUIRED rather than an optimisation because
        #   'no two adjacent' below is unsatisfiable once k > ceil(n/2)
        #   (Sec. II A, p. 5)
    # the cascade then spans the (n-m)-hole sector, which is H_{n,m} relabelled;
    #   C(n,m) = C(n,n-m), so every count below is unchanged
apply X to k of the n qubits, no two adjacent

# --- cascade A gates to span H_{n,m} = {|s>: s_i in {0,1}, sum_i s_i = m} -----
#     dim(H_{n,m}) = C(n,m)                                            (Eq. 1)
repeat
    first layer:   A(theta,phi) on every adjacent pair with an X on one side
    second layer:  A(theta,phi) linking each newly-touched qubit to a free
                       neighbor, until all adjacent qubits are connected
until  C(n,m) A gates placed
    # arrangement is NOT proved for general n,m -- confirmed only by fidelity
    #   sweeps over n,m in {2..6}x{1..5}, stated as a conjecture beyond that
fix phi on the last two A gates placed
    # naive 2*C(n,m) -> minimal 2*C(n,m) - 2 = 2*dim(H_{n,m}) - 2 real parameters

# --- optional: restrict to real coefficients (time-reversal symmetry) --------
if time-reversal required:
    set phi = 0 in every A gate
    fix one theta to zero
    # strips every coefficient's phase, then halves the count to dim(H_{n,m}) - 1

# --- optional: fix s_z (qubits 1..n/2 spin-up, n/2+1..n spin-down) ------------
if s_z required:
    zero every A gate bridging qubit n/2 and n/2+1 across the whole cascade,
        theta = phi = 0
        # each repeat of the primitive pattern can place its own such gate --
        #   the paper's n=4,m=2 example needs four parameters zeroed this way
    keep all of them out of the first layer
        # swap the first/second layer order if one would otherwise land there

return  U(theta,phi) |0>^(x)n
# spans H_{n,m}, or its time-reversal / s_z-restricted subspace, exactly, at
#   the minimal parameter count for whichever symmetries are imposed
# a hand-tuned circuit for one specific small (n,m) can beat this on CNOTs,
#   at the cost of non-nearest-neighbor connectivity (Fig. 4 vs Fig. 3)

Gard, Zhu, Barron, Mayhall, Economou and Barnes cost this family in variational parameters and CNOT gates only. Under Jordan-Wigner, $n$ spin-orbitals with $m$ electrons give a particle-number sector of dimension $\binom{n}{m}$, and absent any other symmetry $2\binom{n}{m}-2$ real parameters is the minimum that spans it; time reversal — eigenstates choosable real, no applied magnetic field — halves that to $\binom{n}{m}-1$. Their linear nearest-neighbour circuit uses $\binom{n}{m}$ two-qubit $A$ gates of three CNOTs each, so at most $3\binom{n}{m}$; dropping gates made redundant by the fixed input $|0\rangle^{\otimes n}$, Eq. (3) gives $3\binom{n}{m}-3m+1$ for $0<m<n/2$, a separate branch at $m=n/2$ and its particle-hole mirror above. The $\binom{n}{m}$ arrangement is a conjecture verified numerically only for $n\le 6$, the authors call this CNOT count "neither unique nor optimal", and parameters still grow exponentially. Imposing $s$ and $s_z$ (even $n$ only) leaves $\dim(H_{n,m,s,s_z})-1$ free angles, with Eq. (4) the subspace size. No shot, measurement, Hamiltonian-term or optimiser-iteration count appears anywhere in the paper or its supplement.

• Symmetry-preserving VQE ansatz · Qiskit

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  The ansatz is constrained to preserve selected particle-number, parity, or spin symmetries.

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• Symmetry-preserving VQE ansatz

  The ansatz is constrained to preserve selected particle-number, parity, or spin symmetries.