We deform the Carlen--Maas--Wirth framework for noncommutative dynamical optimal transport by an Arnold--Nielsen type complexity operator. A positive state-independent operator $G$ compatible with the Hilbert bimodule structure of a noncommutative differential calculus $\partial\colon\M\to\Hcal$ can be absorbed into the calculus itself, \[ \partial_G:=G^{1/2}\partial. \] The corresponding complexity-weighted transport problem is exactly the unweighted transport problem generated by $\partial_G$, whenever the deformed quadratic form remains Dirichlet. In finite dimensions we prove existence of minimizers for density-dependent Petz-class metrics and for fixed physical complexity weights, the latter without commutation between $G$ and the state-dependent mobility. On unitary orbits we identify the induced distance with a quotient metric coming from a right-invariant complexity geometry. This yields an exact Bell-state preparation result via Clairaut's relation and an exactly computed restricted-path upper bound for GHZ preparation; the Lindblad detailed-balance case is included only as entropy-gradient-flow background.
We deform the Carlen--Maas--Wirth framework for noncommutative dynamical optimal transport by an Arnold--Nielsen type complexity operator. A positive state-independent operator G compatible with the Hilbert bimodule structure of a noncommutative differential calculus ∂:\M→\Hcal can be absorbed into the calculus itself, [ \partial_G:=G^{1/2}\partial. ] The corresponding complexity-weighted transport problem is exactly the unweighted transport problem generated by ∂G, whenever the deformed quadratic form remains Dirichlet. In finite dimensions we prove existence of minimizers for density-dependent Petz-class metrics and for fixed physical complexity weights, the latter without commutation between G and the state-dependent mobility. On unitary orbits we identify the induced distance with a quotient metric coming from a right-invariant complexity geometry. This yields an exact Bell-state preparation result via Clairaut's relation and an exactly computed restricted-path upper bound for GHZ preparation; the Lindblad detailed-balance case is included only as entropy-gradient-flow background.
Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra sp(4,R). For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are G0=[ℏ2(u−1)]−1 and G2=[ℏ2(3u−1)]−1, where u=(2ν/ℏ)2. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every u; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives G0/G2=1+2(β/α)u/(u−1); matching Bures throughout the isotropic family would require the state-dependent relation β/α=1/u. At the Bures-Fisher determinant crossing u=φ, pointwise matching is possible only by inserting β/α=φ−1. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.
This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group—often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen—and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds. In a previous paper my collaborators and I—inspired by the UV/IR decoupling that happens in the phenomenon of renormalization—conjectured that high- dimensional metrics that look very different at short scales will often nevertheless give rise at long scales to the same emergent effective geometry. The results of this paper provide evidence for those conjectures, since many complexity metrics that have radically different penalty factors and therefore radically different short- distance properties are shown to belong to the same long-distance equivalence class.
We establish a direct correspondence between Krylov and Nielsen complexity by choosing the Krylov basis to be part of the elementary gate set of Nielsen geometry and selecting a Nielsen complexity metric compatible with the Krylov metric. Up to normalization, the Krylov complexity of a Hermitian operator then equals the length squared of a straight-line trajectory on the manifold of unitaries that connects the identity operator with a precursor operator. The corresponding length provides an upper bound on Nielsen complexity that saturates whenever the straight line is a minimal geodesic. While for general systems we can only establish saturation in the limit of small precursors, we provide evidence that in broad classes of models and for suitable initial operators there is a precise correspondence between Krylov complexity and (the square of) Nielsen complexity for a finite range of precursors.
Overview: This paper develops the state-to-geometry interface of the OT–GKSL architecture for open quantum sources. Its starting point is a familiar low-energy tension: classical general relativity couples spacetime geometry to matter through the Einstein equations, whereas quantum matter may be open, coherent, correlated, preparation dependent, or subject to measurement and environmental records. Semiclassical gravity and stochastic-gravity approaches have long made clear that the relation between quantum sources and classical geometry is not conceptually trivial (Page & Geilker, 1981; Kuo & Ford, 1993; Hu & Verdaguer, 2008). The OT–GKSL architecture addresses this problem by not assuming classical spacetime as part of the native quantum state description. Instead, it separates a persistent quantum/open-state domain from later classical descendants. At the native level, states belong to a faithful finite-dimensional working stratum, ρ ∈ D°(H_eff), and evolve under a declared GKSL dynamics, dρξ/dξ = L*(ρξ). The completely positive Markovian structure follows the standard theory of Gorini, Kossakowski & Sudarshan (1976) and Lindblad (1976). On a faithful reversible detailed-balance component, the dissipative generator admits the noncommutative optimal-transport gradient representation developed in the quantum Markov setting by Carlen & Maas (2017): L*_db(ρ) = - grad_gOT D(ρ || π). Here g_OT is a positive state-space metric and D(ρ || π) = Tr[ρ(log ρ - log π)] is the relative entropy to a faithful stationary state π. The detailed-balance entropy production is σ_db(ρ) = ||grad_gOT D(ρ || π)||²_gOT ≥ 0. This identity concerns the detailed-balance dissipative component, not automatically the complete GKSL trajectory. If L* = L_coh + L_db + L*_rem, then the full relative-entropy balance retains the coherent and remainder contributions separately. The native evolution parameter ξ, the accumulated entropic ordering variable t_ent, and physical readout time t_ro are therefore distinct: ξ ≠ t_ent ≠ t_ro. A physical clock requires a stabilized and calibrated clock record. Its identification with the temporal sector of the Lorentzian spacetime readout requires the additional clock–coframe and proper-time compatibility conditions of the certified spacetime branch. From records to geometric readout: Classical geometry is not identified directly with the native OT geometry. The architecture first constructs physical records. Protocol-complete certified records determine event identity, while a separate geometric coarse-graining determines which event distinctions are used for localization. A 2 rank-D family then supplies local geometric coordinates. On the observed spacetime branch, D = 4. native states → certified physical records → event classes → geometric coarse-graining → rank-D coordinate records → horizontal readout sector → section and soldering →Lorentzian coframe → classical metricThe important distinction is complete event-record content ≠ rank-D coordinate content.Additional records may encode matter properties, preparation data, measurement outcomes, orientation, gauge information, clock information, or internal structure without becoming additional spacetime dimensions. On a regular geometric patch, the coordinate map defines a vertical sector and its OT-orthogonalhorizontal complement, V = ker DΠ_coord, H = V^(⊥ gOT). A local section selects native representatives and a calibrated soldering map transfers the readablehorizontal data to the physical frame bundle. The physical coframe e^a then defines g_ro = η_ab e^a ⊗ e^b, with η_ab = diag(-1,+1,+1,+1)on the observed Lorentzian branch. The positive metric g_OT and the Lorentzian metric g_ro are therefore different mathematical objects with different domains and physical roles. Lorentzian signature is not obtained by changing the signature of g_OT; it enters through the calibrated readout branch.This distinction is compatible with the broader operational idea that physical reference frames and records may themselves be quantum systems rather than external classical structures (Bartlett, Rudolph & Spekkens, 2007; Giacomini, Castro-Ruiz & Brukner, 2019), while the specific record-to-geometry ordering is a commitment of the OT-GKSL architecture Horizontal connection and holonomy bridge: A central result of the paper is the typed separation between native state-space geometry and readout geometry. Let ∇^OT denote the Levi-Civita connection of the applicable OT metric. The connection relevant for the geometric bridge is the projected connection on the horizontal bundle, ∇^H_X Y = P_H ∇^OT_X Y.Its curvature R^H is not generally the restriction of the ambient OT curvature R^OT. In particular, R^OT ≠ 0 does not by itself imply R^H ≠ 0.The readout bridge therefore transports the horizontal connection, rather than silently identifying ambient OT curvature with spacetime curvature. With a local section s and soldering map Ω, the transferred connection has the standard inhomogeneous transformation structure, Γ_B,s = Ω (s*Γ_H) Ω^(-1) - (dΩ) Ω^(-1).It may then be compared with the independently constructed readout connection. The corresponding bridge defect is controlled on the declared certified domain. For a fixed loop C, a connection-level defect gives a length-type transport estimate of Duhamel form. An area-type small-loop estimate requires the stronger input of curvature control, a suitable small contractible loop, and compatible gauge/trivialization data. The paper therefore distinguishes four statements: ambient OT curvature; horizontal curvature and holonomy; transferred/readout holonomy; detector-visible holonomic response. None is identified with the next without an explicit bridge or visibility condition.3. This separation is related conceptually to earlier geometric treatments of mixed-state parallel transport and quantum holonomy (Uhlmann, 1986; Andersson & Heydari, 2014), although the OT-GKSL horizontal/readout bridge is an architecture-specific construction. Finite holonomy and preparation sensitivity. The consolidated formulation distinguishes a nontrivial finite loop signal from its sensitivity to preparation. A finite holonomy may satisfy H(ρ*) ≠ I while nevertheless Dρ H(ρ*) = 0. Thus anontrivial background holonomy does not prove that the holonomy varies under the preparation tested by an experiment.Two operational observables are accordingly kept distinct.Y_H^fin = L_H^fin(H - I) is a finite-loop detector. A body-fixed or Maurer-Cartan response may instead use Ξ_H = L̂_H(H^(-1) ∇_prep H).These are distinct calibrated observables. They coincide only under an explicitly declared translation/calibration relation. Preparation differentiation must also be gauge controlled, by freezing a physical record-defined frame, differentiating a gauge-invariant functional, or using a covariant derivative.Source-side constitutive branch The source/matter descendant is constructed independently of the geometric descendant. Within the Dirac/HS-NJL working class considered in the paper, an internal operator Y defines the positive quadratic tracer κ_Y = Tr(Y†Y). A frozen calibration maps this quantity to an operational coordinate p_κ. The scalar source-side constitutive law is written μ = μ(p_κ), with local susceptibility β_μ(p_κ) = - ∂_(pκ) ln μ(p_κ). A certification gate may determine whether the working point remains admissible, but its derivative is not part of the physical constitutive susceptibility in the certified interior. The scalar constitutive channel and the holonomic channel are mathematically distinct and possess different experimental symmetry handles. Einstein lock: The observed gravitational branch is constrained by an Einstein lock. The two-derivative gravitational kinetic block is fixed to the Einstein-Hilbert form,S_EH[g_ro; G0] = (c³ / 16πG0) ∫ d⁴x sqrt(-g_ro) R[g_ro], with G0 = constant.This is a branch-selection and recovery condition of the OT-GKSL corpus. It is not claimed to be the unique consequence of GKSL dynamics alone. The architecture forbids a readable preparation-dependent factor multiplying the Einstein-Hilbert curvature term on the locked branch. State dependence is instead carried by independently specified source, response, gauge, interface, readout, or higher-order sectors. The gravitational comparison is performed only after geometry and source data have beenindependently fixed: 4 C_E = G_μν[g_ro] - κ0 T^eff_μν - Δ^if_μν. A gravitational claim requires the residual C_E to satisfy its declared norm and tolerance. TheBianchi identity acts as a compatibility condition on these already constructed sectors; it does not define the source after the fact. The branch-resolved Trilemma: The central result is a local and conditional separation theorem for the visible constitutive and holonomic branches. On a finite-dimensional admissible test sector, define the kinematic analysis using the unweighted source coordinate and the holonomic preparation response, h → (dp_κ[h], ζ_H[h]). The constitutive coefficient β_μ is applied only after this kinematic conditioning. Consequently, testing β_μ,* = 0 does not make the underlying analysis map singular by definition. For a normed visible output space, the declared first-order closure has the form DO_vis[h] = N_E(β_μ,* dp_κ[h]) + N_H(ζ_H[h]) + B_oth[h] + R_aud[h]. Here B_oth contains independently constructed visible branches not removed by the frozen protocol, and R_aud is the declared audit remainder. The factorization of the scalar constitutive term through β_μ is an explicit visible/source closure assumption; it is not derived solely from OT variational calculus. At a working point where β_μ,* = 0, if, for a tested variation h, ||N_H(ζ_H[h])|| > ||B_oth[h] + R_aud[h]||, then DO_vis[h] ≠ 0. The final implication follows from the reverse triangle inequality. The nontrivial content lies upstream: construction of the regular record/geometric quotien
See also: Covariant Coherent-Generator Geometry over Quantum Wasserstein State Space This paper arises from the broader OT-GKSL architecture, in which detailed-balance GKSL dynamics are organized through a sequence of transport layers: a current representation, the induced state-space Onsager/quantum-Wasserstein geometry, and subsequent reduction to collective variables. The present work isolates and develops the native-to-collective coarse-graining layer of that architecture as a mathematically self-contained problem. OT-GKSL Pedagogical Guide, Foundations Audit, Uniqueness & Substitutability Analysis, Literature Comparison, and Reading Guidance : A Pedagogical Guide to Understanding the OT–GKSL Architecture. Abstract / Summary: This work studies finite-dimensional coarse-graining of detailed-balance quantum Markov dynamics equipped with a specified Carlen–Maas noncommutative Wasserstein/Onsager geometry. The starting point is a smooth manifold of faithful density matrices 𝓜₊ and a positive Onsager co-metric: K_ρ : T*₊𝓜₊ → T_ρ𝓜₊ with inverse transport metric: g_ρ(u, v) = ⟨K_ρ⁻¹ u, v⟩ For the declared detailed-balance sector, the dissipative drift is assumed to satisfy the Carlen–Maas gradient-flow identity: b_db(ρ) = −K_ρ DF(ρ) where F(ρ) = D(ρ || π) is the Umegaki relative entropy with respect to a faithful stationary state π. A specified current representation is written as: K_ρ = Γ_ρ M_ρ Γ_ρ* where Γ_ρ is the continuity/divergence map and M_ρ is a positive current mobility. For a smooth collective map q : 𝓜₊ → Q, write P_ρ = dq_ρ. At a fixed microscopic representative ρ, the pointwise quotient Onsager co-metric is: K_ρᵠ = P_ρ K_ρ P_ρ* with inverse G_ρᵠ = (K_ρᵠ)⁻¹, and minimum-norm horizontal lift: H_ρ = K_ρ P_ρ* G_ρᵠ The current-to-state kinetic minimization and the minimum-norm Riemannian quotient construction are standard ingredients and are not claimed as new. Their typed composition gives the local collective quadratic contraction: inf_{j : P_ρ[a(ρ) + Γ_ρ j] = dq/dt} 𝓛_cur(ρ, j) = (1/4) ⟨G_ρᵠ(dq/dt − ℓ_ρ), dq/dt − ℓ_ρ⟩ where ℓ_ρ = P_ρ b(ρ). Key Findings & Core Structure: The main purpose of the paper is to separate three issues that remain after this pointwise contraction: 1. Drift Basicity and Mobility Basicity are Distinct: Exact autonomous deterministic closure requires: P_ρ b(ρ) = Lᵇ(q(ρ)) for a vector field Lᵇ defined intrinsically on the collective space. By contrast, an intrinsic section-independent quadratic transport geometry requires: P_ρ K_ρ P_ρ* = Kᵇ(q(ρ)) These are logically independent projectability conditions. In particular, an exact deterministic reduced equation may exist even when the reduced transport cost still depends on the microscopic representative. A reversible three-state Markov-chain example makes this independence explicit. For q = p₁ + p₂, the projected deterministic dynamics closes exactly as: dq/dt = 1 − 2q whereas the logarithmic-mean quotient mobility is: K_pᵇ = (1/4) Λ(4p₁, 2p₃) + (1/4) Λ(4p₂, 2p₃) and therefore depends on the hidden split between p₁ and p₂. Here, Λ(a, b) = (a − b) / (log a − log b) for a ≠ b, with Λ(a, a) = a. 2. Exact Quotient-versus-Section Defect Identity: Let s : U ⊂ Q → 𝓜₊ be a smooth representative section satisfying q ∘ s = Id. Define the section-dependent reduced drift Ls(q) = dq{s(q)} b(s(q)), the horizontal lift H_s, the vertical section tilt V_s = ds − H_s, and the native tangency residual R_s(q) = b(s(q)) − ds_q L_s(q). Both V_s and R_s are vertical. The central exact decomposition is: (1/4) || dsq (dq/dt) − b(s(q)) ||²{g_{s(q)}} = (1/4) ⟨G_sᵠ (dq/dt − L_s), dq/dt − L_s⟩ + (1/4) || V_s (dq/dt − Ls) − Rs ||²{g{s(q)}} The first term is the pointwise quotient cost at the chosen representative. The second term is a nonnegative vertical square and is exactly the additional cost generated by imposing the reconstruction section instead of using the minimum-norm quotient lift. The corresponding metric identity is: s* g = G_sᵠ + V_s* g V_s ≥ G_sᵠ Thus pullback along a parametrized reconstruction and quotient minimization over hidden tangent directions are distinct operations. Horizontal sections satisfy V_s = 0, but such sections need not exist locally: the transport-horizontal distribution may fail the Frobenius integrability condition because of nonzero Ehresmann curvature. 3. Approximate Basicity: Exact fiber basicity is restrictive. The paper therefore introduces representative-dependent drift and mobility defects: ε_b(ρ) = || ℓ_ρ − L̄(q) ||_h ε_K(ρ) = || Kρᵠ − K̄(q) ||{op, h} relative to a reference reduced pair (L̄, K̄) and an auxiliary Riemannian norm h. Assuming uniform ellipticity with constant m > 0, and writing R = || dq/dt − L̄(q) ||_h, the local representative-dependent quadratic cost C_ρ obeys: | C_ρ(q, dq/dt) − C̄(q, dq/dt) | ≤ (1/4) [ (ε_K(ρ) / m²) (R + ε_b(ρ))² + (ε_b(ρ) / m) (2R + ε_b(ρ)) ] Hence the exact quotient theory is recovered as the zero-defect limit, while weak variation of projected drift and mobility along fibers yields an explicit quantitative coarse-graining estimate. Fiber oscillations can additionally be controlled by vertical regularity and the metric diameter of the fiber. Quantum Consistency Check: A two-level Davies generator is linearized around a faithful equilibrium state. In Hilbert–Schmidt tangent coordinates the entropy Hessian and Onsager operator take the diagonal form: Cπ = diag(c⊥, c_⊥, c_z) Kπ = diag(k⊥, k_⊥, k_z) with k_⊥ = Γ₂ / c_⊥ and k_z = 2 Γ₁ p_g p_e. For a tilted local section s_η(z) = (η z, 0, z), the section-defect theorem gives: (1/4) || dsη (dz/dt) − Adb s_η ||²{g_π} = (dz/dt + Γ₁ z)² / (4 k_z) + η² (dz/dt + Γ₂ z)² / (4 k⊥) This example is intentionally local and linearized. It is a noncommutative tangent-space consistency check, not a nonlinear quantum coarse-graining theorem. Novelty and Scope: The paper does not claim novelty for: The Carlen–Maas current-to-state kinetic-energy minimum; Minimum-norm horizontal lifts or Riemannian quotient metrics; The formula K_ρᵠ = dq_ρ K_ρ dq_ρ* in isolation; Reduced Onsager operators as a general concept. The contribution is the assembled reduction framework consisting of: A typed current-to-state-to-collective contraction; The explicit separation between drift projectability and mobility projectability; The exact arbitrary-section vertical-defect identity involving both section tilt and native tangency residual; An explicit perturbative stability theorem for approximate basicity; Geometric control of fiber defects through vertical regularity and fiber diameter. The underlying quotient and section arguments are finite-dimensional Onsager/Riemannian statements once a positive co-metric K_ρ is supplied. Their quantum content in this work comes from their specialization to a declared Carlen–Maas noncommutative transport structure and detailed-balance quantum Markov dynamics. Explicit Limitations: The quadratic current functional introduced in the paper is not claimed to be a microscopic quantum path-space large-deviation rate function. The paper does not prove that an arbitrary physical coarse-graining produces a reduced quantum Markov semigroup whose intrinsic Carlen–Maas Onsager operator is exactly dq_ρ K_ρ dq_ρ*. Establishing such a functorial reduction theorem, for example for conditional expectations onto quantum subalgebras, is identified as a major open problem. Classification & Mathematics Subject Classification (2020): Subjects: Mathematical Physics, Quantum Information / Open Quantum Systems, Optimal Transport, Differential Geometry, Probability / Markov Semigroups, Model Reduction MSC 2020: 81S22, 47D07, 49Q22, 53C21, 60J27 Key References for Independent Validation: E. A. Carlen and J. Maas, Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, Journal of Functional Analysis 273 (2017), 1810–1869. DOI: 10.1016/j.jfa.2017.05.003 E. A. Carlen and J. Maas, Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems, Journal of Statistical Physics 178 (2020), 319–378. DOI: 10.1007/s10955-019-02434-w J. Maas, Gradient flows of the entropy for finite Markov chains, Journal of Functional Analysis 261 (2011), 2250–2292. DOI: 10.1016/j.jfa.2011.06.009 J. Maas and A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Journal of Statistical Physics 181 (2020), 2257–2303. DOI: 10.1007/s10955-020-02663-4 M. Liero, A. Mielke, O. Tse, and J.-J. Zhu, Evolution of Gaussians in the Hellinger–Kantorovich–Boltzmann gradient flow, Communications on Pure and Applied Analysis 31 (2026), 166–198. DOI: 10.3934/cpaa.2025105 S. Becker and W. Li, Quantum statistical learning via quantum Wasserstein natural gradient, Journal of Statistical Physics 182 (2021), Article 7. DOI: 10.1007/s10955-020-02682-1 D. F. Hornshaw, Quantum optimal transport for approximately finite-dimensional C-algebras*, arXiv:1910.03312. 1. Foundations of the Architecture: Foundations |GKSL/Lindblad ; Carlen–Maas ; Jacobson ; Sakharov ; Donoghue ; Lovelock) Establishes the core Einstein-locked OT/GKSL architecture for certified geometric readout and coherence-dependent gravitational sourcing. Optimal-Transport Gravity Trilemma | Identifies the certified operational boundary of geometric readout by proving the fundamental trade-off between temporal resolution, coframe stability, and bridge fidelity. 2. Emergence and Recovery of Classical Physics: Exact Reduced OT/GKSL Equations | Mori–Zwanzig/projection operators ; effective field theory ; Carlen–Maas ; Wilsonian reduction / Demonstrates the controlled recovery of classical Newtonian and gravitational sectors as exact non-linear red
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many of the basic geometric objects associated to this space, including the Levi-Civita connection, the geodesic equation, the curvature, and the Jacobi equation. We show that the optimal Hamiltonian evolution for synthesis of a desired unitary necessarily obeys a simple universal geodesic equation. As a consequence, once the initial value of the Hamiltonian is set, subsequent changes to the Hamiltonian are completely determined by the geodesic equation. We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation. This deformation procedure is illustrated using some three-qubit numerical examples. We study the computational complexity of evaluating distances on Riemmanian manifolds, and show that no efficient classical algorithm for this problem exists, subject to the assumption that good pseudorandom generators exist. Finally, we develop a canonical extension procedure for unitary operations which allows ancilla qubits to be incorporated into the geometric approach to quantum computing.