Symmetry-preserving state-preparation circuits

What it fills

• Choose a parameterised trial state

  Fix the gate structure of a circuit family and leave its rotation angles open. What comes back is not a circuit but the set of states the later optimisation is allowed to search — which is why this is a slot of its own and not a paragraph in one method's write-up.

When it applies

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".

Requires

Every step this method names moves its route along, so there is nothing it needs alongside them.

Example

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)

Cost, as the source states it

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.

Implementations

Nobody has written one up yet. That is a gap in this record, not a statement that the method has never been run — the paper register already records, per paper, which sources report numerics or a hardware run.

What it needs

Nobody has taken this apart yet. That is a gap in this graph, not a claim that the method has no parts.

Other ways to fill the same slot

Different approaches

• Unitary coupled-cluster singles and doubles

  Build the trial state from single and double excitations out of a reference determinant, exponentiated as a unitary. The family is chosen for chemistry rather than for the machine: the state you want is in it by construction, and the circuit that reaches it is deep.

• Hardware-efficient ansatz

  Build the trial state out of the gates and couplings the machine already has, and accept whatever states that reaches. The circuit is shallow because nothing in it was chosen for the chemistry; there is correspondingly no argument that the state you want is inside the family.

• ADAPT-VQE adaptive ansatz

  Refuse to fix the structure in advance. Start from nothing and add one operator at a time, choosing each from a pool according to what the molecule itself indicates, until the energy stops improving. The circuit ends up short because nothing was included that the problem did not ask for.

• qubit-ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  The same grow-it-one-operator-at-a-time construction, with the pool rebuilt out of qubit operators rather than fermionic excitations so that the circuits it produces are shallow enough for near-term hardware.

• k-UpCCGSD ansatz

  Take k repetitions of paired double excitations together with generalized singles, instead of the full set of doubles. The point of the restriction is that the depth then grows linearly in the number of orbitals rather than polynomially, and k is the dial that buys accuracy back.

• Qubit coupled-cluster ansatz

  Skip the fermionic layer and build the ansatz directly in qubit space, ranking candidate entangling operators by how much each would move the energy and keeping the ones that earn their place.

• Particle-hole coupled-cluster circuits

  Rewrite the Hamiltonian around the reference determinant so that what the circuit has to describe is excitations out of it, then build the family from gates that move an electron without creating or destroying one. Staying inside the right particle-number sector is a property of the gates, not something the optimiser has to discover.

• Orbital-optimized coupled-cluster circuits

  Let the orbitals move too. The usual family fixes a basis and varies the amplitudes; this one varies the molecular orbital coefficients alongside them, so the same accuracy is reachable from a smaller active space and a shallower circuit — and the energy becomes fully variational, which is what makes forces available.

• TETRIS-ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  Keep ADAPT's habit of growing the ansatz from measured gradients, and stop adding one operator per round. Several operators acting on disjoint qubits can go in together, filling the same layer instead of stacking — the same circuit, packed rather than piled.

• Iterative qubit coupled cluster a narrower version of Qubit coupled-cluster ansatz

  Stop growing the circuit and grow the Hamiltonian instead. Each round folds the entanglers found so far into the operator by a canonical transformation, so every round runs a circuit of the same size — the cost moves off the device and into the number of terms that have to be measured.

• Generalized singles and doubles ansatz

  Drop the rule that an excitation has to move an electron from an occupied orbital into an empty one. Every pair of orbitals may be coupled, so the circuit stops depending on which reference determinant it was built around — a wider variational manifold, paid for in parameters.

• Batched ADAPT-VQE ansatz a narrower version of ADAPT-VQE adaptive ansatz

  Keep ADAPT's habit of growing the ansatz from measured gradients, and stop adding exactly one operator per round. Every operator whose gradient is close to the largest goes in together, so the ansatz reaches the same size in far fewer rounds — and it is the rounds, not the operators, that cost measurements.

In the Atlas

• Symmetry-preserving VQE ansatz

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

Sources

• Efficient Symmetry-Preserving State Preparation Circuits for the Variational Quantum Eigensolver Algorithm↗Bryan T. Gard, Linghua Zhu, George S. Barron, Nicholas J. Mayhall, Sophia E. Economou, Edwin Barnes · 2019theory yes · simulation yes · hardware no