Qudit-Basis Universal Quantum Computation using χ^{(2)} Interactions
QQudit-Basis Universal Quantum Computation using χ (2) Interactions
Murphy Yuezhen Niu,
1, 2
Isaac L. Chuang,
1, 2, 3 and Jeffrey H. Shapiro
1, 3 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Department of Electrical Engineering and Computer Science,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: March 13, 2018)We prove that universal quantum computation can be realized—using only linear optics and χ (2) (three-wavemixing) interactions—in any ( n + 1) -dimensional qudit basis of the n -pump-photon subspace. First, we exhibita strictly universal gate set for the qubit basis in the one-pump-photon subspace. Next, we demonstrate qutrit-basis universality by proving that χ (2) Hamiltonians and photon-number operators generate the full u (3) Liealgebra in the two-pump-photon subspace, and showing how the qutrit controlled- Z gate can be implementedwith only linear optics and χ (2) interactions. We then use proof by induction to obtain our general qudit result.Our induction proof relies on coherent photon injection/subtraction, a technique enabled by χ (2) interactionbetween the encoding modes and ancillary modes. Finally, we show that coherent photon injection is more thana conceptual tool in that it offers a route to preparing high-photon-number Fock states from single-photon Fockstates. Introduction. — Photons are promising information carri-ers for quantum computers owing to photons’ long room-temperature coherence time, high transmission speed, andhigh-fidelity preparation schemes [1–4], plus the availabilityof efficient photodetectors [5, 6], and the scalable on-chipintegration of linear and nonlinear optical components [7–10]. Architectures for optics-based quantum computationhave gone through dramatic developments over the past twodecades [11–16], but significant obstacles remain to be over-come.Optics-based quantum computation depends on photon-photon interactions for the realization of a universal gate set.The lowest order photon-photon interactions are described byunitary transformations of the form ˆ U = exp( − i ˆ L ) that aregenerated by general two-wave mixing Hamiltonians, ˆ L ∈ { ( g ˆ a ˆ b + g ∗ ˆ a † ˆ b † ) , ( g ˆ a ˆ b † + g ∗ ˆ a † ˆ b ) } , (1)where g is a c -number and ˆ a † and ˆ b † are photon-creation op-erators from different optical modes, so that [ˆ a, ˆ b † ] = 0 , orthe same optical mode, for which [ˆ a, ˆ b † ] = 1 . Unitary trans-formations of this form can realize universal single-qubit ro-tations in the Fock-state basis but are not universal for quan-tum computation without some additional resource. To im-plement universal optics-based quantum computation, four-wave mixing (a χ (3) interaction) was previously consideredto be the lowest-order optical nonlinearity that will suffice inthis regard [13, 17, 18]. The inherent weakness of χ (3) in-teractions, however, has precluded their delivering the high-fidelity gates required to make optics-based quantum compu-tation practical [19–21]. Linear-optical quantum computation(LOQC) [14, 22–24] circumvents the need for photon-photoninteractions through postselection, but this approach comeswith the need for a prohibitive number of perfect single-photon ancillae to cope with LOQC’s probabilistic nature andthe ubiquitous photon loss [15, 25–27].One way to circumvent the weakness of photon-photon in-teractions is to employ the lowest -order nonlinearity that can provide universal quantum computation, viz., the χ (2) inter-action whose three-wave-mixing Hamiltonians can be decom-posed into linear combinations of the following terms [28] ˆ G = iκ (cid:104) ˆ a † s ˆ a † i ˆ a p − ˆ a s ˆ a i ˆ a † p (cid:105) , ˆ G = κ (cid:104) ˆ a † s ˆ a † i ˆ a p + ˆ a s ˆ a i ˆ a † p (cid:105) . (2)Here, { ˆ a † k : k = s, i, p } are the photon-creation operators ofthe interaction’s signal, idler, and pump modes, and the real-valued κ quantifies the interaction’s strength.The efficiencies of χ (2) interactions have been steadily im-proving over the past decade [29–42]. Moreover, owing to theimportance of χ (2) interactions in quantum state transductionfor superconducting and ion-trap qubits, the platforms of in-terest for χ (2) interactions have expanded beyond traditionalnonlinear crystals [36–41], bringing full utilization of theirquantum dynamics closer to reality.Coherent photon conversion, i.e., χ (2) interactions definedin (2) in which the signal, idler, and pump modes are allquantum mechanical, was first proposed by Koshino [43],and later used by Langford et al . [42] to show how univer-sal quantum computation can be realized with that resource inthe single-photon qubit basis. We refer to such interactionsas full-quantum χ (2) interactions, to distinguish them from pumped χ (2) interactions, in which a nondepleting coherent-state pump reduces (2) to the two-wave interactions shown in(1). Langford et al .’s groundbreaking work, however, is notwithout drawback. Available schemes for correcting photonloss [11, 44–46], viz., the dominant error in photonic quan-tum computation, require either measurement-based or χ (3) gates on the encoded basis. Thus Ref. [42] does not provide a χ (2) approach that facilitates photonic quantum computationthat is robust to photon loss.In this Letter, and its companion paper [47], we showhow the work of Langford et al . can be extended to amore natural computational basis for χ (2) -based quantumcomputation in which photon-loss errors can be addressed. a r X i v : . [ qu a n t - ph ] M a r More generally, we prove that χ (2) interactions plus linearoptics can provide a strictly universal gate set for quantumcomputation in any ( n + 1) -dimensional qudit basis of the n -pump-photon subspace. Because any d -qudit unitary gatecan be described by a Lie group element of SU (( n + 1) d ) ,the universality of a given class of Hamiltonians is directlyrelated to that class’s Lie algebra and the Lie group it gener-ates via the exponential map [48]. Thus we use Lie-algebraanalysis to identify code subspaces that are closed under χ (2) Hamiltonian evolutions [50, 51]. Our Lie-algebra analysisunderlies the symmetry-operator formulation of qudit-basiserror-correcting codes for photon-loss errors and the universalgate-set constructions in the encoded basis that we reportin [47]. Hence our proposal provides a χ (2) approach tophotonic quantum computation that is robust to photon loss.We begin the development of our universality results with asummary of the linear optics and the χ (2) resources we shallemploy. We follow with qubit and qutrit universality proofs,as preludes to our induction proof for the general qudit case. Linear Optics and χ (2) Resources. — The linear opticsresources we require are readily available: dichroic mir-rors and phase shifters. The pumped χ (2) resource we re-quire is quantum-state frequency conversion (QFC) [52–54],which converts a frequency- ω in single-photon Fock state to afrequency- ω out single-photon Fock state. The full-quantum χ (2) resources we require are: second-harmonic generation(SHG), which converts a frequency- ω in two-photon Fock stateto a frequency- ω in single-photon Fock state; type-I phase-matched spontaneous parametric downconversion (SPDC),which converts a frequency- ω in single-photon Fock state toa frequency- ω in two-photon Fock state; and generalized sum-frequency generation (SFG θ ), which accomplishes the statetransformation [55]SFG θ | , , (cid:105) = cos( θ ) | , , (cid:105) + sin( θ ) | , , (cid:105) , (3)where | n s , n i , n p (cid:105) denotes a three-mode Fock state containing n s frequency- ω s photons, n i frequency- ω i idler photons, and n p frequency- ω p pump photons, with the pump’s frequencysatisfying ω p = ω s + ω i . Universality in the Qubit Basis. — The Lie group generatedby χ (2) Hamiltonian evolutions is a subgroup of the unitarygroup U , hence it is compact. A compact Lie group, togetherwith its generating Lie algebra, are completely reducible.This means that they can be written as a direct sum of irre-ducible representations over the state space H ≡ ⊕ ∞ n =1 H n ,whose irreducible subspaces, {H n } , are labeled by their pumpmode’s maximum photon number n , i.e., they are the n -pump-photon subspaces spanned by the three-mode Fock-state bases {| , , n (cid:105) , | , , n − (cid:105) , . . . , | n, n, (cid:105)} . For qubit universality,we therefore encode in the one-pump-photon subspace H ,using the three-mode Fock states, | ˜0 (cid:105) = | , , (cid:105) , | ˜1 (cid:105) = | , , (cid:105) , (4) for our logical-qubit basis states. Here, the signal and idler areboth at frequency ω with orthogonal polarizations, the pumpis at frequency ω , and all three share a common spatial mode.Universality is proved by the following theorem. Theorem 1 . Universal quantum computation can be realizedwith χ (2) interactions and linear optics in any qubit basis ofthe one-pump photon subspace. Proof : The χ (2) Hamiltonians, ˆ G and ˆ G , defined in (2)are proportional to the Pauli ˆ Y and Pauli ˆ X operators in thelogical-qubit basis, which are universal for realizing single-qubit rotations. So, to complete our χ (2) universality proof forthe logical-qubit basis in (4), it suffices for us to show that wecan construct a controlled- Z qubit gate for that basis [12], i.e.,a gate (denoted Λ [ Z ] in what follows) that imparts a π -radphase shift to the | ˜1 (cid:105) c | ˜1 (cid:105) t component of the joint state of thecontrol (subscript c ) and target (subscript t ) qubits. Moreover,because Λ [ Z ] can be sandwiched between single-qubit χ (2) rotations to achieve the controlled- Z function in any H qubitbasis, Theorem 1 will be proved once we have established howto realize Λ [ Z ] .Figure 1 shows our optical circuit [56] for the Λ [ Z ] gatefor the logical-qubit basis in (4). The control and target qubitsenter on the upper and lower rails, respectively. QFC1 shiftsthe frequency of control qubit’s pump photon (if present) from ω to ω (cid:48) , so that dichroic mirrors (DMs) are able to directpump photons from the control and target qubits to the cen-ter rail’s SFG π gate, where they serve as modes 1 (frequency ω ≡ ω (cid:48) ) and 2 (frequency ω = 2 ω ). This gate impartsa π -rad phase shift if and only if pump photons are presentfrom both the control and target qubits. Thus, after anotherset of DMs restore the control and target pump photons to thetop and bottom rails, respectively, the Λ [ Z ] gate—and hencethe proof of Theorem 1—is completed by QFC2, which shiftsthe frequency of the control qubit’s pump photon (if present)from ω (cid:48) to ω . Note that each χ (2) element Fig. 1 acts ononly one of its potentially excited bosonic-mode inputs, e.g.,QFC1 affects its pump-mode input but neither its signal-modeinput nor its idler-mode input. Such modal selectivity putsa burden on experimental realization. In particular, QFC1and QFC2 will require a different nonlinear medium than willSFG π . This difficulty, however, may disappear once high-efficiency nondepleted χ (3) induced χ (2) interactions becomeavailable [33, 34, 42]. target DM DMDM DMDM DM
SFG π QFC1 QFC2 control
FIG. 1. Schematic for constructing the Λ [ Z ] gate in the logical-qubit basis (4) using χ (2) interactions and linear optics. QFC1 andQFC2: quantum-state frequency conversions. DM: dichroic mirror.SFG π : generalized sum-frequency generation (3) with θ = π . Universality in the Qutrit Basis. — For qutrit universality,we encode in H using the three-mode Fock states | ˜0 (cid:105) = | , , (cid:105) , | ˜1 (cid:105) = | , , (cid:105) , | ˜2 (cid:105) = | , , (cid:105) , (5)for our logical-qutrit basis states. Here, the signal and idlerhave frequency ω and are orthogonally polarized, while thepump has frequency ω , and all three share a common spa-tial mode. These states can be prepared by type-II phase-matched SPDC in the two-pump-photon subspace [35], andare naturally confined to this subspace under χ (2) interac-tions. It follows that restricting linear combinations of the χ (2) Hamiltonians, ˆ G , ˆ G , the modal photon-number opera-tors, { ˆ N k ≡ ˆ a † k ˆ a k : k = s, i, p } , and the nested commutatorsof these operators to the two-pump-photon subspace H con-stitutes a Lie algebra g . The Lie group H associated with g is found from the exponential map exp : g → H , wherefor each group element ˆ h ∈ H , ∃ ˆ E ∈ g and t ∈ R suchthat ˆ h = exp( it ˆ E ) . For simplicity, in all that follows, we set κ = 1 in the Hamiltonians ˆ G and ˆ G . We begin our univer-sality demonstration with a theorem about g . Theorem 2 . The Lie algebra g is u (3) . Proof : First we prove that u (3) ⊆ g . From the origi-nal χ (2) Hamiltonians ˆ G and ˆ G , we can obtain all trans-formations generated by linear combinations of ˆ G , ˆ G , ˆ N s , ˆ N i , ˆ N p and their nested commutators. Using the vector v T ≡ [ v v v ] to represent the qutrit | ψ (cid:105) = v | , , (cid:105) + v | , , (cid:105) + v | , , (cid:105) , we obtain the matrix representations ˆ G = i (cid:104) ˆ a † s ˆ a † i ˆ a p − ˆ a s ˆ a i ˆ a † p (cid:105) = − i √ − −√ , (6) ˆ G = 12 (cid:104) ˆ a † s ˆ a † i ˆ a p + ˆ a s ˆ a i ˆ a † p (cid:105) = 12 √
22 0 0 √ , (7) ˆ G = i [ ˆ G , ˆ G ] = − , (8) ˆ G = i [ ˆ G , ˆ G ] = 3 , (9) ˆ G = i [ ˆ G , ˆ G ] = 3 i − , (10) ˆ G = 12 (cid:16) i [ ˆ G , ˆ G ] + i [ ˆ G , ˆ G ] (cid:17) = 34 , (11) ˆ G = i [ ˆ G , ˆ G ] = 3 i −
10 1 0 , (12) ˆ G = 12 (1 − ˆ N p ) = 12 − , (13) ˆ G = 12 (cid:32) ˆ N s + ˆ N i N p (cid:33) = , (14)for all the independent generators, where the second equali-ties apply in the two-pump-photon subspace H . It is thenstraightforward to verify that the Gell-Mann matrices arisingfrom linear combinations of the above generators are: ˆ λ = ˆ G / , ˆ λ = − ˆ G / , (15) ˆ λ = 2 ˆ G + ˆ G , ˆ λ = √
2( ˆ G − ˆ G / , (16) ˆ λ = √
2( ˆ G − ˆ G / , ˆ λ = 4 ˆ G / , (17) ˆ λ = 4 ˆ G / , ˆ λ = ( ˆ G + 6 ˆ G ) / √ . (18)Gell-Mann matrices are one representation of the complete setof linearly independent generators for the su (3) Lie algebra.Together with ˆ G they form the complete set of generators for u (3) , proving that u (3) ⊆ g . We complete our proof of Theorem 2 by showing that g ⊆ u (3) . Because the two-pump-photon subspace H is closedunder g , every Lie group element ˆ h ∈ H is generated by an ˆ E ∈ g via ˆ h = exp( it ˆ E ) for some t ∈ R . As exp( it ˆ E ) is aunitary transformation in the two-pump-photon subspace, wehave H ⊂ U (3) . Furthermore, this condition holds if and onlyif g ⊆ u (3) , thus finishing Theorem 2’s proof.Refs. [57–59] show that if operators { ˆ G k } and their nestedcommutators generate the Lie algebra u (3 m ) , then they canbe used to construct a universal set of unitaries U k ( t ) =exp( − it ˆ G k ) in the m -qutrit subspace. Setting m = 1 wehave the following claim. Claim 1 . Universal single-qutrit rotations can be realizedwith χ (2) interactions.Universal qutrit computation entails not only universalsingle-qutrit unitary gates but also universal two-qutrit unitarytransformations in H ⊗ , so we need the following theorem. Theorem 3 . Universal qutrit quantum computation can berealized with χ (2) interactions and linear optics in any qutritbasis of the two-pump-photon subspace. Proof : From Claim 1 we know that arbitrary U (3) qutritrotations can be realized with χ (2) interactions. It is alsoknown [12, 60–62] that a universal single-qutrit gate set plus acontrolled- Z gate for the logical-qutrit basis in (5)—denoted Λ [ Z ] —are universal for qutrit computation in any qutrit basisof the two-pump-photon subspace H .The Λ [ Z ] gate realizes the unitary transformation Λ [ Z ] | ˜ j (cid:105) c | ˜ k (cid:105) t = ( − δ ˜ j ˜2 δ ˜ k ˜2 | ˜ j (cid:105) c | ˜ k (cid:105) t for states in H , where δ uv is the Kronecker delta. Figure 2 shows how this gatecan be realized using χ (2) interactions and linear optics. Thecontrol and target qubits enter on the upper and lower rails,respectively, where second-harmonic generators (SHGs) con-vert two-photon Fock-state pumps at frequency ω to a single-photon Fock state at frequency ω . The shaded block labeled Λ [ Z ] is the same gate shown in Fig. 1 except that: (1) itsQFC1 converts a frequency- ω single-photon Fock state to afrequency ω (cid:48) single-photon Fock state; (2) its first set of DMsroute the frequency- ω (cid:48) photon (if present) from the upper rail SHG
SPDCSPDC
DM DMDM DM
QFC1
DM DM
SFG π SHG
QFC2 ⇤ [ Z ] control target FIG. 2. Schematic for constructing the Λ [ Z ] gate in the logical-qutrit basis (5) using χ (2) interactions and linear optics. SHG:second-harmonic generation; Λ [ Z ] : the optical circuit from Fig. (1)with modifications described in the text. SPDC: type-I phase-matched spontaneous parametric downconversion. and the frequency- ω photon (if present) from the lower railto the SFG π block on the center rail; (3) its SFG π block is ar-ranged to apply a π -rad phase shift to the state | , , (cid:105) , whosefirst two entries are the photon numbers of its frequency- ω (cid:48) and frequency- ω inputs; (4) its second set of DMs re-turn the frequency- ω (cid:48) and frequency- ω photons to the up-per and lower rails, respectively; and (5) its QFC2 convertsa frequency- ω (cid:48) single-photon Fock state to a frequency- ω single-photon Fock state. The SPDC blocks then complete the Λ [ Z ] gate—by converting frequency- ω single-photon Fockstates (if present) to frequency- ω two-photon Fock states—because the Λ [ Z ] block has imparted a π -rad phase shift tothe | ˜2 (cid:105) c | ˜2 (cid:105) t component of the original input state. Togetherwith Claim 1, the Λ [ Z ] construction proves Theorem 3. Universality in the ( n +1) -Dimensional Qudit Basis. — Theculmination of our χ (2) universality work is the following the-orem. Theorem 4.
Universal qudit quantum computation can berealized with χ (2) interactions and linear optics in any ( n +1) -dimensional basis of the n -pump-photon subspace. Proof : Our proof is by induction. We have already shownthat Theorem 4 holds for n = 1 and n = 2 . The inductionproof is completed by assuming that Theorem 4 holds for n = m , and then showing that it holds for n = m + 1 .The details appear in [55]. Here we just note that theyinvolve a Lie-group result [51, 63] and coherent photoninjection/subtraction. Coherent photon injection/subtractionare full-quantum χ (2) interactions between the encodedmodes and ancillary modes. Although used as a conceptualtool in the proof of Theorem 4, coherent photon injectionhas independent merit owing to its enabling prepara-tion of high-photon-number Fock states from single-photonFock states. Thus we devote the next section to its description. Coherent Photon Injection. — The coherent photon injec-tion used in our universality proof is a generalization of aresult from Hubel et al . [64]. To illustrate how it works,suppose we start with the qubit-basis state | ˜1 (cid:105) = | , , (cid:105) from (4) with the goal of generating the qutrit-basis state | ˜2 (cid:105) = | , , (cid:105) from (5). Coherent photon injection accom- plishes this task as follows. We adjoin the | , , (cid:105) systemwith an ancillary pump mode (photon creation operator ˆ a † p (cid:48) )that has the same frequency as, but is orthogonally polarizedto, the pump mode of | , , (cid:105) . We then turn on the χ (2) in-teraction ˆ G a = (cid:104) ˆ a † s ˆ a † i ˆ a p (cid:48) + ˆ a s ˆ a i ˆ a † p (cid:48) (cid:105) between the originalsignal-idler modes and the ancillary pump mode to realize thetransformation e iπ ˆ G a / | , , (cid:105)| (cid:105) a = | , , (cid:105)| (cid:105) a . This co-herent photon injection has transformed the qubit-basis state | ˜1 (cid:105) = | , , (cid:105) in the one-pump-photon subspace to the qutrit-basis state | ˜1 (cid:105) = | , , (cid:105) in the two-pump-photon subspace.A qutrit-basis χ (2) gate can now rotate | ˜1 (cid:105) to | ˜2 (cid:105) in the two-pump-photon subspace [55]. Insofar as the pump mode is con-cerned, this overall procedure has converted a single-photonFock-state input to a two-photon Fock-state output. The in-jection process can now be repeated to transform | , , (cid:105) to | , , (cid:105) , after which a χ (2) -enabled rotation in the three-pump-photon subspace will yield | , , (cid:105) . In this manner,high-photon-number Fock states can be prepared using onlysingle-photon sources and full-quantum χ (2) interactions. Conclusions. — We have shown that universal optics-basedquantum computation using only linear optics and χ (2) inter-actions is possible in any ( n + 1) -dimensional qudit basis ofthe n -pump-photon subspace, with the natural basis being thethree-mode Fock states {| , , n (cid:105) , | , , n − (cid:105) , . . . , | , , n (cid:105)} of frequency- ω , orthogonally-polarized signal and idlermodes, and a frequency- ω pump mode, all of which sharea common spatial mode. Our work extends the usual gate-model universality to the universality of χ (2) Hamiltonianinteractions in their irreducible subspaces. Such extensionfacilitates error correction for photon loss by providing asymmetry-operator formalism for hardware-efficient quantumerror correction [47]. Moreover, Lie algebraic understandingof χ (2) interactions opens a path for defining an Abelian groupthat would enable fault-tolerant quantum computation that isrobust to photon loss and physical rotation errors. To reachthe end of that path, however, will require technology devel-opment.The resources required for our qudit-basis χ (2) quantumcomputation are: single photon sources, and linear optics, plus χ (2) interactions. High-quality linear optics (dichroic mirrorsand phase shifters) are already available, and high-efficiencyquantum-state frequency conversion (the pumped χ (2) inter-action we need) have been demonstrated. But, because cur-rently available or demonstrated single-photon sources andfull-quantum χ (2) (SHG, SFG π , and SPDC) interactions fallshort of what our architectures require, continued advancesin these technologies must occur before our quantum compu-tation proposals become practical. There is some reason foroptimism in this regard, e.g., the efficiency of the χ (2) non-linearity has been improved from − [42] to − [65] inless than a decade. Furthermore, state-of-the-art experimen-tal realizations of strong χ (2) interactions—including in solid-state circuits [36], flux-driven Josephson parametric ampli-fiers [37, 38, 41], superconducting resonator arrays [39, 40],nondepleted four-wave-mixing-induced three-wave mixing inphotonic microstructured fibers [33, 34, 42], χ (2) interac-tions inside ring resonators [66], and nonlinear interactions infrequency-degenerate double-lambda systems [67]—are clos-ing the gap between theory and practical applications of full-quantum χ (2) interactions.M. Y. N. and J. H. S. acknowledge support from Air ForceOffice of Scientific Research Grant No. FA9550-14-1-0052.M. Y. N. acknowledges support from the Claude E. ShannonResearch Assistantship. I. L. C. acknowledges support fromthe National Science Foundation Center for Ultracold Atoms.M. Y. N. acknowledges early discussion with B. C. Sanderson Lie-group analysis of χ (2) interactions. [1] Z. Yuan, B. E. Kardynal, R. M. Stevenson, A. J. 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Comput. , 773 (2009). Supplemental Material
Here we present more information about the generalized sum-frequency generation (SFG θ ) transformation, the proof of themain paper’s Theorem 4, and the details of the | , , (cid:105) → | , , (cid:105) transformation needed to complete the coherent photoninjection process for generating a two-photon Fock state from two single-photon Fock states. GENERALIZED SUM-FREQUENCY GENERATION
Consider the closed t ≥ joint-state evolution of the signal, idler, and pump modes in the main paper’s one-pump-photonsubspace, Span {| , , (cid:105) , | , , (cid:105)} , under the action of the χ (2) Hamiltonian i (cid:126) κ (ˆ a † s ˆ a † i ˆ a p − ˆ a s ˆ a i ˆ a † p ) , where κ is real valued.From [35, 42] we have that this joint state satisfies | ψ ( t ) (cid:105) = v ( t ) | , , (cid:105) + v ( t ) | , , (cid:105) , for t ≥ , (19)where ˙ v ( t ) = − κv ( t ) , ˙ v ( t ) = κv ( t ) , for t ≥ . (20)When | ψ (0) (cid:105) = | , , (cid:105) , we find that | ψ ( t ) (cid:105) = cos( κt ) | , , (cid:105) + sin( κt ) | , , (cid:105) , for t ≥ . (21)Setting θ = κt then gives us the generalized sum-frequency generation (SFG θ ) transformation. PROOF OF THE MAIN PAPER’S THEOREM 4
To complete the induction proof of the main paper’s Theorem 4, we must show that χ (2) interactions and linear opticsare universal in the ( n + 1) -pump-photon subspace H n +1 , given that these resources are universal in all j -pump-photonsubspaces, H j , for ≤ j ≤ n . Our proof has three steps, which make use of H n +1 ’s decomposition into its bulk states, H (cid:48) n +1 ≡ Span {| , , n (cid:105) , | , , n − (cid:105) , . . . , | n, n, (cid:105)} , and its boundary states, H (cid:48)⊥ n +1 ≡ Span {| , , n + 1 (cid:105) , | n + 1 , n + 1 , (cid:105)} .First, under the proof’s premise, we prove the universality of χ (2) interactions and linear optics in the bulk-state subspace H (cid:48) n +1 . This step relies on the assumption that χ (2) interactions and linear optics permit coherent photon subtraction/injection tobe performed. Second, we show how to achieve universality that spans H n +1 by including the boundary states, H (cid:48)⊥ n +1 , for thatsubspace. Finally, we show that χ (2) interactions suffice to realize the coherent photon subtraction/injection operations used inStep 1. Step 1 : We begin by introducing additional χ (2) Hamiltonians between signal and idler modes with an ancillary pump mode, ˆ G ,p (cid:48) = i (cid:104) ˆ a † s ˆ a † i ˆ a p (cid:48) − ˆ a s ˆ a i ˆ a † p (cid:48) (cid:105) , (22) ˆ G ,p (cid:48) = 12 (cid:104) ˆ a † s ˆ a † i ˆ a p (cid:48) + ˆ a s ˆ a i ˆ a † p (cid:48) (cid:105) , (23)as well as between ancillary signal and idler modes and the pump mode, ˆ G ,s (cid:48) i (cid:48) = i (cid:104) ˆ a † s (cid:48) ˆ a † i (cid:48) ˆ a p − ˆ a s (cid:48) ˆ a i (cid:48) ˆ a † p (cid:105) , (24) ˆ G ,s (cid:48) i (cid:48) = 12 (cid:104) ˆ a † s (cid:48) ˆ a † i (cid:48) ˆ a p + ˆ a s (cid:48) ˆ a i (cid:48) ˆ a † p (cid:105) . (25)Here we have taken κ = 1 , and we have assumed that: (1) the ancillary pump mode and the pump mode have the same ω frequency, but are orthogonally polarized; and (2) the ancillary signal and idler modes have frequencies ω s (cid:48) = ω + ∆ ω and ω i (cid:48) = ω − ∆ ω , so that they are orthogonal to each other and to the signal and idler modes, which both have frequency ω .Next, we assume that linear optics plus the χ (2) interactions ˆ G ,p (cid:48) and ˆ G ,p (cid:48) allow us to realize the unitary transformation ˆ U sip (cid:48) that accomplishes the following coherent photon subtraction on H n +1 , ˆ U sip (cid:48) | j, j, n + 1 − j (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) = | j − , j − , n + 1 − j (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) , for ≤ j ≤ n, n ≥ . (26)Likewise, we assume that linear optics plus the χ (2) interactions ˆ G ,s (cid:48) i (cid:48) and ˆ G ,s (cid:48) i (cid:48) allow us to realize the unitary transformation ˆ U ps (cid:48) i (cid:48) that accomplishes the following coherent photon subtraction on H n +1 , ˆ U ps (cid:48) i (cid:48) | j − , j − , n + 1 − j (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) = | j − , j − , n − j (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) , for ≤ j ≤ n, n ≥ . (27)Concatenating these transformations then converts any state in H (cid:48) n +1 into a corresponding state in H n − by means of the basistransformation ˆ U ps (cid:48) i (cid:48) ˆ U sip (cid:48) | j, j, n + 1 − j (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) = | j − , j − , n − j (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) , for ≤ j ≤ n, n ≥ . (28)This concatenation, however, will also unavoidably impact the H n +1 boundary basis states | , , n + 1 (cid:105) and | n + 1 , n + 1 , (cid:105) ,i.e., there will be { c k } and { d k } such that ˆ U ps (cid:48) i (cid:48) ˆ U sip (cid:48) | , , n + 1 (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) = n +1 (cid:88) k =0 c k | , , n + 1 − k (cid:105)| k, k (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) , (29) ˆ U ps (cid:48) i (cid:48) ˆ U sip (cid:48) | n + 1 , n + 1 , (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) = n +1 (cid:88) k =0 d k | n + 1 − k, n + 1 − k, (cid:105)| , (cid:105) s (cid:48) i (cid:48) | n − k (cid:105) p (cid:48) . (30)Note that the transformations in Eqs. (29) and (30) do not contain any states with a | , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) component, which isin the ( n − -pump-photon subspace of the ancillary signal, idler, and pump modes. We can thus realize any gate ˆ U target n thatacts on the bulk-state subspace H (cid:48) n +1 , without affecting the boundary basis states, using only linear optics and χ (2) interactions.In particular, let ˆ˜ U target n be the mapping of ˆ U target n to a unitary that acts on H n − . Then, from Theorem 4’s premise, we knowwe can realize the controlled unitary gate Λ[ ˆ˜ U target n ] that is conditioned on the ancillary modes’ being in their | , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) state: Λ[ ˆ˜ U target n ] = ˆ˜ U target n ⊗ | , (cid:105) s (cid:48) i (cid:48) | n − (cid:105) p (cid:48) p (cid:48) (cid:104) n − | s (cid:48) i (cid:48) (cid:104) , | + ˆ I n − ⊗ (cid:88) { j,l }(cid:54) = { ,n − } | j, j (cid:105) s (cid:48) i (cid:48) | l (cid:105) p (cid:48) p (cid:48) (cid:104) l | s (cid:48) i (cid:48) (cid:104) j, j | , (31)where ˆ I n − is the H n − identity operator. We then undo the effects in Eqs. (29) and (30) by conjugating with ˆ U ps (cid:48) i (cid:48) ˆ U sip (cid:48) , andso obtain ˆ U target n = ( ˆ U ps (cid:48) i (cid:48) ˆ U sip (cid:48) ) † Λ[ ˆ˜ U target n ] ˆ U ps (cid:48) i (cid:48) ˆ U sip (cid:48) . (32)This result completes Step 1, because χ (2) interactions and linear optics enable realization of the coherent photon injectionoperations embodied by ( ˆ U ps (cid:48) i (cid:48) ˆ U sip (cid:48) ) † as they are the inverses of the coherent photon subtraction operations in ˆ U ps (cid:48) i (cid:48) ˆ U sip (cid:48) ,whose realizations with those resources we have already assumed (and will prove in Step 3). Step 2 : We have shown that universal gates in H (cid:48) n +1 can be implemented from χ (2) interactions and linear optics, given thatsuch resources suffice to realize universal gates in H j for ≤ j ≤ n and to do coherent photon subtraction and injection. It thusremains for us to show that this universality result can be extended to H n +1 under that same premise, i.e., we must now includethe boundary states, H (cid:48)⊥ n +1 , that are in H n +1 but not in H (cid:48) n +1 . To do so we start by decomposing the restrictions to H n +1 of themain paper’s χ (2) Hamiltonians ˆ G and ˆ G into their components that act within H (cid:48) n +1 and those that act between the boundarystates, H (cid:48)⊥ n +1 , and the bulk states, H (cid:48) n +1 . Using ˆ˜ G and ˆ˜ G to denote the H n +1 -restricted Hamiltonians, and assuming κ = 1 ,we can show that ˆ˜ G = 12 (cid:34) √ n + 1 ˆ σ yn +1 ,n + ( n + 1)ˆ σ y , + n − (cid:88) k =1 ( n + 1 − k ) √ k + 1 ˆ σ yk +1 ,k (cid:35) , (33) ˆ˜ G = 12 (cid:34) √ n + 1 ˆ σ xn +1 ,n + ( n + 1)ˆ σ x , + n − (cid:88) k =1 ( n + 1 − k ) √ k + 1 ˆ σ xk +1 ,k (cid:35) , (34)where ˆ σ xk +1 ,k and ˆ σ yk +1 ,k are the Pauli ˆ X and ˆ Y operators between the basis states | n − k, n − k, k + 1 (cid:105) and | n + 1 − k, n + 1 − k, k (cid:105) .Next, from Step 1, we know that we can implement the H (cid:48) n +1 unitary gates ˆ U ( θ ) =exp (cid:104) − iθ (cid:16)(cid:80) n − k =1 ( n + 1 − k ) √ k + 1 ˆ σ yk +1 ,k (cid:17)(cid:105) and ˆ U ( θ ) = exp (cid:104) − iθ (cid:16)(cid:80) n − k =1 ( n + 1 − k ) √ k + 1 ˆ σ xk +1 ,k (cid:17)(cid:105) with onlythe presumed resources. Then, in order to implement ˆ˜ G and ˆ˜ G using only two-dimensional subspaces, we leverage the Trotterformula to obtain the following entangling operators between H (cid:48) n +1 and the two boundary basis states | n − k, n − k, k + 1 (cid:105) and | n + 1 − k, n + 1 − k, k (cid:105) : ˆ V ( θ ) = exp (cid:2) iθ ( √ n + 1 ˆ σ yn +1 ,n + ( n + 1)ˆ σ y , ) (cid:3) = exp (cid:34) iθ ˆ G − iθ (cid:32) n − (cid:88) k =1 ( n + 1 − k ) √ k + 1 ˆ σ yk +1 ,k (cid:33)(cid:35) = lim m →∞ (cid:104) e iθ ˆ G /m ˆ U ( θ ) /m (cid:105) m , (35) ˆ V ( θ ) = exp (cid:2) iθ ( √ n + 1 ˆ σ xn +1 ,n + ( n + 1)ˆ σ x , ) (cid:3) = exp (cid:34) iθ ˆ G − iθ (cid:32) n − (cid:88) k =1 ( n + 1 − k ) √ k + 1 ˆ σ xk +1 ,k (cid:33)(cid:35) = lim m →∞ (cid:104) e iθ ˆ G /m ˆ U ( θ ) /m (cid:105) m . (36)At this point, we can choose rotation angles such that ˆ V ( θ ) ˆ V ( θ ) and ˆ V ( θ ) ˆ V ( θ ) respectively construct individualPauli ˆ X and ˆ Y rotations either between | n + 1 , n + 1 , (cid:105) and | n, n, (cid:105) , or between | , , n + 1 (cid:105) and | , , n (cid:105) . Six of theserotations, with different angles, can then be used to realize any SU(2) rotation between | , , n + 1 (cid:105) and | , , n (cid:105) , or between | n + 1 , n + 1 , (cid:105) and | n, n, (cid:105) [51, 63, 68, 69]. So, because any unitary transformation can be decomposed into products ofunitary transformations between two neighboring basis states of any chosen order [69], we have proven that χ (2) interactionsand linear optics are sufficient to realize universal gates in H n +1 assuming that these resources suffice to realize such gatesin H j for ≤ j ≤ n , and that they also suffice for realizing the coherent photon subtraction/injection operations that wereemployed in Step 1. Step 3 : With Step 2 in hand, completing Theorem 4’s induction proof only requires showing that χ (2) interactions can be usedto implement the coherent photon subtraction operations— ˆ U sip (cid:48) and ˆ U s (cid:48) i (cid:48) p from Eqs. (26) and (27)—that were employed inStep 1. (Step 1 also used the coherent photon injection operations ˆ U † sip (cid:48) and ˆ U † s (cid:48) i (cid:48) p , but their realizations are merely inverses ofthe unitaries for their associated subtraction processes.) In particular, we need only demonstrate that the available resources areuniversal in the joint Hilbert space H joint n ≡ (cid:0) ⊕ nj =1 H (cid:48) j (cid:1) ⊗ C s (cid:48) i (cid:48) ⊗ C p (cid:48) = Span {| j, j, k (cid:105)| l, l (cid:105) s (cid:48) i (cid:48) | h (cid:105) p (cid:48) : 1 ≤ j + h ≤ n, ≤ k + l ≤ n, h, l ∈ { , }} .Let U sum be the unitary group in ⊕ nj =1 H (cid:48) j , and let S be the unitary group generated by linear optics and χ (2) interactions. Bythe induction proof’s premise we know that U sum ⊂ S . Moreover, because linear optics alone is universal in C s (cid:48) i (cid:48) , and so too isit universal in C p (cid:48) , we have that U sum ⊗ U s (cid:48) i (cid:48) ⊗ U p (cid:48) ⊂ S , where U s (cid:48) i (cid:48) is the unitary group in C s (cid:48) i (cid:48) and U p (cid:48) is the unitary groupin C p (cid:48) . The proof from Ref. [70] will now establish the universality of S in H joint n if we can show there are imprimitive gates ˆ V s (cid:48) i (cid:48) and ˆ V p (cid:48) in S that perform entangling operations between subspaces ⊕ nj =1 H (cid:48) j and C s (cid:48) i (cid:48) and between ⊕ nj =1 H (cid:48) j and C p (cid:48) , viz.,for | ψ (cid:105) n ∈ ⊕ nj =1 H (cid:48) j , | ψ (cid:48) (cid:105) s (cid:48) i (cid:48) ∈ C s (cid:48) i (cid:48) , and | ψ (cid:48) (cid:105) p (cid:48) ∈ C p (cid:48) , we have that ˆ V s (cid:48) i (cid:48) | ψ (cid:105) n | ψ (cid:48) (cid:105) s (cid:48) i (cid:48) and ˆ V p (cid:48) | ψ (cid:105) n | ψ (cid:48) (cid:105) p (cid:48) are entangled states.Below we will establish the existence of ˆ V p (cid:48) ; a similar argument, which we omit, will do the same for ˆ V s (cid:48) i (cid:48) .Consider the initial product state, | ψ (cid:105) n | ψ (cid:48) (cid:105) p (cid:48) = n − (cid:88) k =1 k − (cid:88) j =0 α j | j, j, k − j (cid:105) (cid:34) (cid:88) q =0 β q | q (cid:105) p (cid:48) (cid:35) , (37)in ( ⊕ nj =1 H (cid:48) j ) ⊗ C p (cid:48) . Based on the universality of χ (2) interactions plus linear optics in j -pump-photon subspaces with ≤ j ≤ n ,there exists a unitary ˆ U (cid:48) sip (cid:48) , generated by ˆ G ,p (cid:48) and ˆ G ,p (cid:48) , that transforms | ψ (cid:105) n | ψ (cid:48) (cid:105) p (cid:48) into ˆ U (cid:48) sip (cid:48) | ψ (cid:105) n | ψ (cid:48) (cid:105) p (cid:48) = n − (cid:88) k =1 k − (cid:88) j =0 ( α j β γ j, + α j +1 β γ j +1 , ) | j + 1 , j + 1 , k − j (cid:105)| (cid:105) p (cid:48) + n − (cid:88) k =1 k − (cid:88) j =1 ( α j β γ j, + α j − β γ j − , ) | j − , j − , k − j (cid:105)| (cid:105) p (cid:48) (38)where the { γ j,q } are determined by the χ (2) Hamiltonian evolution between signal, idler and ancillary pump modes. Equa-tion (38) generates entanglement whenever γ j,q (cid:54) = γ j (cid:48) ,q (cid:48) for any j (cid:54) = j (cid:48) or q (cid:54) = q (cid:48) . Such will always be the case with anappropriate rotation angle θ of the χ (2) Hamiltonian evolution generated by Eqs. (22) and (23). Thus χ (2) interactions and linearoptics are universal in H joint n and hence sufficient to implement coherent photon subtraction/injection, completing the proof ofthe main paper’s Theorem 4. χ (2) -ENABLED APPROACH FOR REALIZING THE | , , (cid:105) → | , , (cid:105) TRANSFORMATION
The main paper’s description of using coherent photon injection to generate a two-photon Fock state from two single-photonFock states via coherent photon injection required a χ (2) -enabled approach to accomplishing the | , , (cid:105) → | , , (cid:105) transfor-mation. Here we will provide a suitable implementation for that transformation. The first step is to apply the χ (2) Hamiltonian ˆ G = i (cid:126) κ (ˆ a † s ˆ a † i ˆ a p − ˆ a s ˆ a i ˆ a † p ) to the | , , (cid:105) state for time t = 2 π/ κ √ . Using the H evolution equation from [35], we find that | ψ (cid:105) ≡ e − i π ˆ G/ κ √ | , , (cid:105) = − | , , (cid:105) − | , , (cid:105) + 1 √ | , , (cid:105) . (39)The second step is to send the preceding quantum state through the optical circuit shown in Fig. 3. Here, the pump mode isrouted to the upper rail by the first dichroic mirror (DM), while the second-harmonic generation (SHG) block converts a two-photon Fock-state idler (if present) to a single-photon Fock-state at the pump frequency. That pump-frequency single-photonstate is converted back into a two-photon Fock-state idler by spontaneous parametric downconversion (SPDC). In that process itaccumulates a π -rad Berry’s phase. The circuit is completed by using a DM to put the pump mode on the output rail. The neteffect of this circuit is then to transform | ψ (cid:105) to | ψ (cid:105) = − | , , (cid:105) − | , , (cid:105) − √ | , , (cid:105) . (40) SHG SPDC
DM DM
FIG. 3. Optical circuit for the second step of the | , , (cid:105) → | , , (cid:105) transformation. DM: dichroic mirror. SHG: second-harmonic generation.SPDC: spontaneous parametric downconversion. The last step is to evolve | ψ (cid:105) under ˆ G for time t = 2 π/ κ √ to obtain | ψ (cid:105) ≡ e − i π ˆ G/ κ √ | ψ (cid:105) = −| ,,
FIG. 3. Optical circuit for the second step of the | , , (cid:105) → | , , (cid:105) transformation. DM: dichroic mirror. SHG: second-harmonic generation.SPDC: spontaneous parametric downconversion. The last step is to evolve | ψ (cid:105) under ˆ G for time t = 2 π/ κ √ to obtain | ψ (cid:105) ≡ e − i π ˆ G/ κ √ | ψ (cid:105) = −| ,, ,,
FIG. 3. Optical circuit for the second step of the | , , (cid:105) → | , , (cid:105) transformation. DM: dichroic mirror. SHG: second-harmonic generation.SPDC: spontaneous parametric downconversion. The last step is to evolve | ψ (cid:105) under ˆ G for time t = 2 π/ κ √ to obtain | ψ (cid:105) ≡ e − i π ˆ G/ κ √ | ψ (cid:105) = −| ,, ,, (cid:105) ,,