Quantum circuit for chemistry simulation

Quantum circuits for chemistry simulation are based on second quantization Hamiltonian coefficients for one-body and two-body interactions. Jordan-Wigner series that conserve parity can be defined so that selected CNOT gates are removed from the associated circuits. Basis change gates such as Hadamard or Y-gates can be coupled to some or all qubits of a quantum circuit or cancelled in view of corresponding gates in adjacent circuits. In some examples, CNOT gates can be moved to different circuit locations.

FIELD

The disclosure pertains to quantum computational systems for evaluation of chemical systems.

BACKGROUND

One application of quantum computing is in the computation of molecular properties that are defined by quantum mechanics. Such quantum computations would have a variety of applications such as in pharmaceutical research and development, biochemistry, and materials science. Conventional computing approaches are suitable for only the simplest quantum chemical computations due to the significant computational resources required. Typical many-body systems of interest generally cannot be evaluated. Quantum chemical computational techniques can require significantly fewer computational resources, and permit computation of the properties of many-body systems.

Current approaches to quantum chemical computations exhibit significant limitations. A standard circuit model uses one and two body Hamiltonian terms. This circuit changes basis and then entangles all of the required qubits, rotates the result, unentangles the qubits, and finally changes back to the original basis. While this conventional approach can produce useful results, very large numbers of gate operations are required, and improved approaches are needed.

SUMMARY

Quantum circuits for chemistry simulation include at least one reduced Jordan-Wigner circuit coupled to a plurality of qubits and include a plurality of CNOT gates corresponding to respective spin orbitals. One or more CNOT gates associated with a full Jordan-Wigner series is omitted or, alternatively, is included in a reduced Hamiltonian circuit coupled to qubits that are associated with the spin orbitals. The reduced Hamiltonian circuit can be based on one body or two body Hamiltonian coefficients associated with a material of interest. In some examples, the circuit includes a plurality of reduced Hamiltonian circuits situated in series and having different basis change gates. In other examples, the circuit includes an output side reduced Jordan-Wigner string corresponding to the input side reduced Jordan-Wigner string, wherein the output side reduced Jordan-Wigner string is situated after the plurality of reduced Hamiltonian circuits. In some examples, the reduced Hamiltonian circuits include Hadamard gates and Y-gates as basis change gates. In other examples, the CNOT gates are coupled to an ancillary or entanglement qubit.

In some examples, methods of defining a quantum circuit associated with at least a selected one-body or two-body Hamiltonian coefficient associated with second quantization comprise defining a reduced Jordan-Wigner string associated with spin orbitals coupled by a Hamiltonian coefficient. A reduced Hamiltonian circuit is defined based on the Hamiltonian coefficient, and the reduced Jordan Wigner string is coupled to the reduced Hamiltonian circuit on an input side. In some examples, CNOT gates of the reduced Jordan Wigner circuit are coupled to an entanglement qubit.

The foregoing and other features, and advantages of the disclosed technology will become more apparent from the following detailed description, which proceeds with reference to the accompanying figures.

DETAILED DESCRIPTION

In some examples, values, procedures, or apparatus' are referred to as “lowest”, “best,” “minimum,” or the like. It will be appreciated that such descriptions are intended to indicate that a selection among many used functional alternatives can be made, and such selections need not be better, smaller, or otherwise preferable to other selections. The term “sequence” is used to describe a series of operations that are executed in series as well as to describe series of quantum circuits or gates that are coupled in series.

Quantum computational methods and apparatus are described herein with respect to quantum gates that are used to implement quantum computations. The quantum gates are representations of physical operations that are to be applied to one or more qubits that can be constructed using various physical systems. In one example, qubit operations (i.e., gates) are applied to qubits defined by states of polarization of optical radiations. Quantum circuits can be defined by a plurality of quantum gates, arranged in a particular order.

For convenience, quantum circuits as described be situated sequentially, first, last, prior to, before, or other terms defining a circuit order from an input to an output.

In the disclosed examples, quantum circuits and methods are disclosed that permit reduction in the complexity of entanglement from O(N) to O(1) within a circuit that defines operations corresponding to a multi-body Hamiltonian. In some example, Hadamard gates are used, wherein the Hadamard gate H is defined as:

H=12⁢(111-1).
A Pauli Y-gate is defined as:

Quantum simulation of molecules can be based on a decomposition of a molecular time evolution operator Û. For small, simple molecules that exhibit substantial symmetries such as the hydrogen molecule, a molecular Hamiltonian can be implemented exactly using relatively small numbers of quantum gates. For more complex molecules, such simple decompositions may be difficult or impossible. Nevertheless, for many chemical systems, suitable exact or approximate decompositions are available.

In a typical method, the molecular chemical Hamiltonian is expressed in second quantized form, and the Jordan-Wigner transformation is used to express terms of the Hamiltonian in a spin ½ representation. A unitary propagator is decomposed into a product of time-evolution operators for non-commuting terms of the molecular Hamiltonian based on a Trotter-Suzuki expansion in which terms associated with non-commutation are neglected. Quantum circuits are defined so as to correspond to each of the time-evolution operators. Using second quantization and Jordan-Wigner transformation, a Hamiltonian is generated that can be represented as qubit operations. Thus, a quantum circuit can implement a quantum algorithm for simulating a time-evolution operator obtained from a molecular Hamiltonian. In the following, such quantum circuits and quantum computational methods are described in more detail.

In a basis having N spin orbitals, a Hamiltonian H associated with one and two body interactions of electrons can be represented as:

H=∑pq⁢tpq⁢cp†⁢cq+12⁢∑pqrq⁢Vpqrs⁢cp†⁢cq†⁢cr⁢cs.
The terms in such a representation can be obtained using classical computing methods.

In many systems of interest, the terms with the largest magnitude are Hppterms. The Hpqqpterms are typically next in magnitude, followed by the Hpqterms, the Hpqqrterms, and finally the Hpqrsterms, but other orderings by magnitude are possible. The Hpqqrand the Hpqterms are generally related as follows after a basis transformation of the single particle states to find a Hartree-Fock ground state:

tpq+∑r⁢Vprrq⁢nr=0,
wherein nr=0,1 is the occupation number in a Hartree-Fock state.

As noted above, quantum chemistry computations can be conveniently expressed in a so-called second quantization form, wherein a Hamiltonian operator H is expressed as:

H=∑m⁢hpq⁢p†⁢q+∑m⁢hpqrs⁢p†⁢q†⁢rs(1)
wherein p, q, r, and s represent spin orbital locations, with each molecular orbital occupied by either a spin-up or spin-down particle, or both or neither. The hpqand hpqrsvalues are the amplitudes associated with such particles and the terms with a dagger (†) correspond to particle creation and terms without a dagger refer to particle annihilation. The hpqand hpqrsvalues can be obtained exactly or estimated. For example using a Hartree-Fock procedure,

The second-quantized Hamiltonian can be mapped to qubits. The logical states of each qubit can be associated with occupancy of a single-electron spin-orbital, wherein 0 denotes occupied, and 1 denotes unoccupied. A system with N single-electron spin-orbitals can be represented with N qubits. Systems with any numbers of electrons up to N can be represented using N qubits. In other representations, larger numbers of qubits can be used.

The Jordan Wigner transformation can be used to transform creation and annihilation operators so as to be represented using Pauli spin matrices. The time-evolution operator may not be readily representable as a sequence gates, but a Hamiltonian expressed as a sum of one and two-electron terms whose time-evolution operators can each be implemented using a sequence of gates. The unitary time evolution operator can be approximated using Trotter-Suzuki relations based on time-evolution of non-commuting operators. For a Hamiltonian

Errors associated with non-commutative operators can be reduced using a Trotter-Suzuki expansion. For example, a time evolution operator for a system associated with non-commutating operators A and B can be expressed as:
eA+B=(eA/neB/n)n,
which is exact for n→∞. In typical quantum computing approaches, each term in a product of exponentials of operators is associated with a corresponding quantum circuit.
Cancellation of Jordan-Wigner Strings

Each Trotter-Suzuki step typically comprises a step implementing a unitary transformation exp(iAδt) controlled by an additional ancilla qubit that is used to perform phase estimation, wherein A is a term in the series expansion of H shown above, and δtis an angle which depends upon the Trotter-Suzuki step. A second quantized basis can be used, with one qubit per spin orbital. For an Hpqterm with p<q, the associated controlled unitary transformation is exp(i(XpXq+YpYq)(Zp+1Zp+2. . . Zq−1)θ wherein a product of Zp+1Zp+2. . . Zq−1implements a Jordan-Wigner string and θ depends upon a coefficient tpq, and on δt. For Hpqrs, p<q<r<s there are several possible controlled unitary transformations, of the form exp(i(XpXqXrXs)(Zp+1Zp+2. . . Zq−1)(Zp+1Zp+2. . . Zs−1)θ) with Jordan-Wigner strings from p+1 to q−1 and from r+1 to s−1. For each p,q,r,s, it may be necessary to implement several of these terms, with XpXqXrXsreplaced by other Pauli operators and combinations thereof, such as XpXqYrYs, etc. . . . , having an even number of Xs and an even number of Ys. These different choices are referred to X, Y basis choices.

For convenient illustration, qubits are labeled and arranged in a sequential order, typically from p to q, but other orderings are possible. In the examples described herein, each spin orbital is associated with a single qubit, but additional qubits can be used.

As used herein, a reduced Jordan-Wigner (or entanglement) sequence applied to p,q,r,s qubits corresponding to spin-orbitals p,q,r,s wherein p,q,r,s are integers such that p<q<r<s, includes CNOT gates coupled to qubits p, . . . s, except for qubits p,q,r,s. Such a sequence is typically used in processing in association with two body (Hpqrs) terms in a second quantization representation of a Hamiltonian. A reduced Jordan-Wigner (or entanglement) sequence applied to p,q qubits corresponding to spin-orbitals p,q wherein p,q are integers such that p<q, includes CNOT gates coupled to qubits p, . . . q, except for qubits p,q. Such a sequence is typically used in processing in association with one body Hpqterms in a second quantization representation of a Hamiltonian. In either case, the Jordan-Wigner string is associated with computing the parity of a given set of qubits.

A reduced (two body) Hamiltonian circuit includes one or more input/output basis change gates (for example, Hadamard gates or Y gates) coupled to p,q,r,s qubits and one or more input/output CNOT gates coupling p,q,r,s qubits. Such input or output CNOT gates can be referred to as interior Jordan-Wigner sequences as they can be provided within the reduced circuit. In some cases, both input side and output side basis change gates are included, or only input or output side basis change gates, or only selected ones of input or output side basis change gates. Such a circuit also includes a controlled rotation gate (a cntol-Z gae) defined by a value of a two body Hamiltonian coefficient of the form Hpqrsand applied to a qubit associated with an s-spin orbital. A reduced (one body) Hamiltonian circuit includes one or more input/output basis change gates (for example, Hadamard gates or Y gates) coupled to p,q qubits and one or more input/output CNOT gates coupling p,q qubits. Such input or output CNOT gates can be referred to as interior Jordan-Wigner sequences as they can be provided within the reduced circuit. In some cases, both input side and output side basis change gates are included, or only input or output side basis change gates, or only selected ones of input or output side basis change gates. Such a circuit also includes a control rotation gate defined by a value of a one body Hamiltonian coefficient of the form Hpqand applied to a qubit associated with an q-spin orbital.

A conventional circuit for implementation of Hpqterms is shown inFIGS. 1A-1B. In the example ofFIGS. 1A-1Band other examples, only selected qubits are shown for convenient illustration. Referring toFIG. 1A, Hadamard gates102A,104A implement a change in basis from state |0> to state (|0>−1>)/√{square root over (2)} for qubits P and Q. A first sequence of CNOT gates106(an entanglement sequence or Jordan-Wigner string) performs Jordan-Wigner entanglement operations coupling qubits P, P+1, . . . , Q, and a controlled rotation circuit108applies a rotation/phase change associated with a quantum computation under consideration to the Q qubit. The circuit108is generally defined based on a Hamiltonian or phase matrix associated with a chemical system of interest, in this case, based on a value of Hpq. A second sequence110of CNOT gates (a Jordan-Wigner string) is applied coupling qubits P, P+1, . . . Q, followed by Hadamard circuits102B,104B that that perform an additional change of basis, returning to an input basis as H2=YY†=I, wherein I is the identity matrix.

Similar processing is also required in a Y-basis. Referring toFIG. 1B, Pauli Y-gates (hereinafter simply Y-gates)122A,122B and124A,124B are situated to implement a change in basis from state |0> to state i|1) for qubits P and Q, respectively. Entanglement sequences (Jordan-Wigner sequences)126,130of CNOT gates couple qubits P, P+1, . . . , Q and are situated before and after and a rotation/phase gate128that is based on a Hamiltonian or phase operator of a quantum system of interest, in this case, based on Hpg

For computations based on Hpq, the circuits ofFIGS. 1A-1Bare repeated in series for each Trotter-Suzuki step. Typically circuits are provide for all combination of p, q, so that if N spin orbitals are used, N2circuits are needed for each Trotter-Suzuki step.

FIGS. 1C-1Dillustrate portions of a quantum circuits140,160that are defined based on two body interactions and implement operations associated with Hpqrsterms. The circuit140ofFIG. 1Cincludes input side Hadamard gates142A-142D coupled to qubits P, Q, R, S, respectively, and a first CNOT entanglement (Jordan-Wigner) sequence146is then applied. The sequence146includes CNOT gates that couple qubits P, P+1, . . . , S but qubits R, Q are directly coupled, and CNOT gates are not included to couple Q and Q+1, Q+1 and Q+2, . . . , R−1, R. A controlled rotation circuit148associated with a Hamiltonian term of interest (i.e., Hpqrs) receives the entangled qubits, and an output side CNOT entanglement sequence150similar to the sequence146is then applied. Output side Hadamard gates144A-144D then implement basis changes to the input bases for qubits P, Q, R, S. The circuit160ofFIG. 1Dincludes input side Y gates162A-162D that implement change of bases for qubits P, Q, R, S, respectively, and a first CNOT entanglement sequence166that situated to be applied after the basis changes and the Jordan Wigner sequence146. A controlled rotation circuit168associated with a Hamiltonian coefficient of interest (Hpqrs) is then coupled to the S qubit. An output side CNOT entanglement sequence160is configured as shown (and similar to the entanglement sequences ofFIG. 1C) and output side Y gates164A-164D are coupled to qubits P, Q, R, S, respectively, to return to the original basis prior to the input side Y gates162A-162D.

The two body circuits ofFIGS. 1C-1Dare similar to the one body circuits ofFIGS. 1A-1Bexcept that four spin-orbitals (p, q, r, s) are included, and there will be 2, 4, 6 or 8 copies of the circuit. As shown inFIGS. 1C-1D, the number of entanglement gates (CNOTs) used grows larger with each successive term. As the number of spin-orbitals grows (N), the number of CNOTs grows with the same order (O(N)). The number of CNOT gates required can be significantly reduced in the disclosed circuits and methods as discussed below. Note also thatFIGS. 1A-1Dshow gate sequences for a single Trotterization step, and for many such steps, application of large numbers of CNOT gates in entanglement sequences can make implementation difficult.

With reference toFIG. 2, an alternative circuit200for computation based on one-body terms Hpqincludes an initial entanglement sequence202of CNOT gates that couple qubits P+1, P+2, . . . , Q−1. An initial control-Z gate204couples a phase qubit and a Q qubit. An H-basis circuit206includes a set207A of Hadamard basis change gates that are coupled to P and Q qubits followed by a CNOT gate209A coupling the P and Q qubits. A control-Z gate211is coupled to the phase qubit and the Q qubit and applies a phase based on the value of Hpqrs. Subsequent to the control-Z gate211, a CNOT gate209B is applied followed by a set207B of Hadamard gates that perform a basis change back to the original basis. A circuit208then follows that includes Y-gates for a basis change along with CNOT gates and a control-Z gate that area similarly situated as the Hadamard gates and CNOT gates209A,209B of circuit206. A control-Z gate210and a final entanglement sequence212are then coupled at the output of the circuit208. The initial entanglement sequence202includes fewer CNOT gates than the conventional circuit ofFIG. 1A, and the entanglement sequences within the circuits206,208couple only the P and Q qubits.

Additional circuits such as circuits206,208can be used for additional Trotter-Suzuki steps. For convenience, circuits such as the circuits206,208are referred to as H-basis circuits and Y-basis circuits, respectively. Although an H-basis circuit precedes a Y-basis circuit inFIG. 2, other orders can be used, for example, Y-basis circuits prior to H-basis circuits. In a series implementing additional Trotter-Suzuki steps, an initial entanglement sequence and an initial control-Z gate are provided, followed by alternating H-basis and Y-basis circuits. A final control-Z gate and a final entanglement sequence follow the alternating H-basis and Y-basis circuits.

Referring toFIG. 3, a circuit300includes an initial Jordan-Wigner sequence302that includes CNOT gates coupling qubits P+1, P+2, . . . , Q−1 followed by a control-Z gate. An H-basis circuit304and a Y-basis circuit (similar to the circuits206,208ofFIG. 2) apply phase/rotation changes based on a value of Hpq. While a final gate sequence308associated with the circuits304,306could be included, along with an initial gate sequence310for the next H-basis, Y-basis sequence, this is unnecessary. The product CNOT2=Z2=I, wherein I is the identity matrix, so that the combination of the sequences308,310is the identity operator, and thus these sequences can be eliminated. Additional Trotter-Suzuki steps are applied by an H-circuit312and a Y-circuit314and additional circuits not shown. A final Y-circuit is followed by a final control-Z/CNOT gate sequence320.

Circuits for processing based on Hpqvalues can be further simplified with addition of an auxiliary qubit referred to for convenience as an “entanglement” qubit. A quantum circuit for computation based on N spin orbitals can then include a phase qubit, computation qubits I, . . . , N, and an entanglement qubit. Portions of an example circuit400are shown inFIG. 4, in which only qubits P, P+1, . . . , Q404are shown along with a phase qubit402and an entanglement qubit406. Qubits408,410are associated with orbitals P and Q, respectively. A set420of CNOT gates couples each of the qubits P+1, . . . ,Q−1 to the entanglement qubit406, and a control-Z gate410couples the entanglement qubit406to the Q qubit. An H-circuit422includes Hadamard gates424,425,426,427coupled to the P and Q qubits for a basis transformations, and CNOT gates430,432that couple the P qubit to the Q qubit. A control-Z gate440is situated between the CNOT gates430,432and applies a phase change/rotation based on the value of Hpq. A Y-circuit452includes Y-gates454,455,456,457coupled to the P and Q qubits for a basis transformations, and CNOT gates460,462that couple the P qubit to the Q qubit. A control-Z gate470is situated between the CNOT gates460,462and applies a phase change/rotation based on the value of Hpq. The circuit400can be used for a plurality of Trotter-Suzuki iterations with additional copies of circuits such as circuits422,452. A final control-Z gate480and a set482of CNOT gates follow a final Y-circuit.

Additional one-body circuits are shown inFIGS. 5A-5C. Referring toFIG. 5A, a circuit includes a set502of CNOT gates coupling qubits P+1, P+2, . . . , Q−1 to an entanglement qubit. A control-Z gate504is followed by an H-basis circuit506and a Y-basis circuit508that include CNOT gates coupling the P, Q qubits, basis change gates (H or Y, respectively), and control rotation gates507,509based on an Hpqvalue. A control-Z gate510coupling the Q and Entanglement qubits follows the Y-basis circuit508, and corresponds to the control Z gate504. Circuit511includes a CNOT gate512coupling the Q qubit to the Entanglement qubit and a control-Z gate coupling the Q+1 qubit and the Entanglement qubit. The CNOT gate512be viewed as an additional gate that is required in the set502for processing based on spin orbitals P, Q+1, i.e. Hp(q+1). While conventionally a mirror image set of CNOT gates corresponding to the set502is provided, this mirror image set combined with a preparatory set for the Hp(q+1)leaves only the CNOT gate512. An H-basis circuit518and Y-basis circuit520for the P, Q+1 include control rotation gates519,521based on a value of Hp(q+1). Circuits for additional Trotter-Suzuki steps, or for other spin orbitals can be included as well. In any case, serially processing qubits P, Q and P, Q+1 requires only single additional CNOT coupling the Q and Entanglement qubits at the transition between P, Q and P, Q+1.

Referring toFIG. 5B, a circuit includes a set530of CNOT gates coupling qubits (P+1, P+2), . . . , (Q−2,Q−1). A control-Z gate532is followed by an H-basis circuit534and a Y-basis circuit536that include CNOT gates coupling the P, Q qubits, basis change gates (H or Y, respectively), and control rotation gates535,537based on an Hpqvalue and coupled to the Q and phase qubits. A control-Z gate540coupling the Q−1 and Q qubits follows the Y-basis circuit508, and corresponds to the control Z gate532. A set542of CNOT gates is illustrated, showing concluding CNOT gates for the P, Q circuits534,536and initial CNOT gates for P, Q+1 H-basis circuit548. However, these CNOT gates can be omitted as all cancel except for CNOT gate543. A control-Z gate544couples the Q and Q+1 qubits and corresponds to the control-Z gate532used for processing based on spin orbitals P, Q. Only a portion of the H-basis circuit548is shown, including a control rotation gate coupling phase and Q+1 qubits, and based on a value of Hp(q+1). Circuits for additional spin orbitals and additional Trotter-Suzuki steps can be provided.

FIG. 5Cillustrates a circuit such as that ofFIG. 5B. A set560of CNOT gates couples qubits (P+1, P+2), . . . , (Q−2, Q−1). A control-Z gate562is followed by an H-basis circuit564and a Y-basis circuit566that include CNOT gates coupling the P, Q qubits, basis change gates (H or Y, respectively), and control rotation gates based on an Hpqvalue and coupled to the Q and phase qubits. A control-Z gate567coupling the Q−1 and Q qubits follows the Y-basis circuit566, and corresponds to the control Z gate562, but for different qubits. A CNOT gate568is followed by a control-Z gate569coupling the Q, Q+1 qubits. The CNOT gate568can be viewed as the sole non-cancelling CNOT gate such as those shown in the542ofFIG. 5B. The control-Z gate569couples the Q and Q+1 qubits and corresponds to the control-Z gate562used for processing based on spin orbitals P, Q. An H-basis circuit570and a Y-basis circuit578for processing based on spin orbitals P, Q+1, i.e., using Hp(q+1)follow the control-Z gate569. Circuits for additional spin orbitals and additional Trotter-Suzuki steps can be provided.

FIGS. 2-5Cillustrate circuits for computation based on values of Hpq. The circuits are simpler, using fewer gates, and permit more rapid computation. However, computations based on Hpqrsvalues generally are more complex and are more time-intensive. A representative circuit for use with these values is illustrated inFIG. 6. Referring toFIG. 6, computation qubits P, P+1, . . . , Q of a set of N qubits are illustrated in a circuit600, wherein N is a number of spin orbitals. For computations associated with Hpqrs, a set602of CNOT gates couples qubits (P+1, P+2), . . . , (Q−2, Q−1) and qubits (R+1, R+2), . . . , (S−2, S−1). A control-Z gate604couples qubits S−1, S. An H-transform circuit606includes Hadamard gates coupled to the qubits P, Q, R, S. A set of CNOT gates608couples qubits (P, Q), (Q, R), and (R, S). A control Z gate610applies a phase change/rotation based on a value of Hpqrs. A set612of CNOT gates is then followed by an H-transform circuit614that includes Hadamard gates associated with P, Q, R, S qubits. For convenience, a circuit630that includes the H transform circuits606,614(the basis change gates, in this example, Hadamard gates), the sets608,612of CNOT gates coupling the P, Q, R, S qubits, and the control-Z gate associated with value of Hpqrsis referred to as an H-basis circuit. Similar circuits can be provided using Y-gates instead of Hadamard gates on some or all of the P, Q, R, S qubits. These circuits can be arranged in series with the H-basis circuit630. The table below shows some possible combinations of basis change gates assigned to different qubits.

QubitPQRSCircuitHHHHHHHHHHYYHHYYYYHHYYHHYYYYYYYY
The sequence of circuits can be repeated for additional Trotter-Suzuki iterations. After the final circuit of the sequence, a control-Z gate620is situated to couple the S−1, S qubits followed by a set622of CNOT gates similar to the set602but in reverse order.

Subterms in a circuit can be re-ordered to cancel Jordan-Wigner strings to obtain a speedup of up to a factor of N.FIGS. 7A-7Bshow circuits701,711for selected basis terms (such as HHYY, YYHH) for Hpqrsand Hpqr′s′, respectively, wherein s≠s′,r≠r′. The circuits701,711include respective Jordan-Wigner strings702,710. The R, S circuit701includes an HHHH circuit704, an HHYY circuit706, a YYHH circuit708, and a YYYY circuit710. The R′, S′ circuit711includes corresponding basic circuits714,715,718, and720.FIGS. 7C-7Dillustrate additional quantum circuits for additional qubits defined by CNOT gate strings722,732and various basis circuits724-730,732-740.

The circuits ofFIGS. 7A-7Dcan be simplified by reordering. Referring toFIG. 8A, an HHYY basis circuit804for Hpqrsis coupled to an HHYY circuit806for Hpqr′s′, instead of coupling HHYY and YYHH basic circuits for Hpqrsin series as shown inFIGS. 7A-7B. Because H2=YY†=I, the final H and Y gates of the circuit804and the initial H and Y gates of the circuit806that operation on the same qubits can be eliminated.FIG. 8Bshows a reduced HHYY basis circuit814for Hpqrsand a reduced HHYY basis circuit816for Hpqr′s′. Initial and/or final basis change gates are removed.

Adding additional qubits such as an entanglement qubit or one or more ancillary qubits allows significant parallelization. Addition of more than one ancilla qubit can increase the extent of parallelization as gates that act on distinct qubits can be executed in parallel. For example, terms associated with Hp′q′r′s′and Hpqr′s′, can be executed in parallel if p<q<r<s<p′<q′<r′<s′ since the associated unitary operators act on different qubits. For two Hpqrsterms, given any choice of p,q,r,s and p′,q′,r′,s′ for which the sites intersect at an even number of sites in the Jordan-Wigner string (i.e., the associated series of CNOT gates) of the p,q,r,s term, the p′,q′,r′,s′ term does not change parity and hence can be moved through the Jordan-Wigner string. This is referred to as nesting as terms can be executed in parallel when one sits inside another (for example, when p<p′<q and r<r′<s).FIGS. 9A-9Dillustrate a circuit prior to nesting and CNOT gate removal. Many CNOT gates in a sequence902of CNOT gates (most of the CNOT gates of a full Jordan-Wigner series) can be cancelled. A result of this cancellation is shown inFIGS. 10A-10Dat1002. A series of CNOT gates (such as the server710ofFIG. 7B, referred to as a Jordan-Wigner string)710can be simplified to a series1002as shown inFIGS. 10A-10D. Further cancellations are possible in1002using the ability to commute CNOTs that act on the same target ancilla. A corresponding nested circuit is shown inFIGS. 11A-11Cusing ancillary qubits1102. Different orderings are possible.

The discussion above applies to arbitrary Hpqand Hpqrs. However, in many applications these will be sparse, meaning that they have many zero entries. Examples of this include simulations of the Hubbard model, simulations of long polymers, and simulations with symmetry. In such cases, many terms are zero, but large reductions in the complexity can still be obtained by the appropriate ordering of the terms for each case.

Representative Computing Environments

FIG. 12and the following discussion are intended to provide a brief, general description of an exemplary computing environment in which the disclosed technology may be implemented. Although not required, the disclosed technology is described in the general context of computer executable instructions, such as program modules, being executed by a personal computer (PC). Generally, program modules include routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types. Moreover, the disclosed technology may be implemented with other computer system configurations, including hand held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, and the like. The disclosed technology may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.

With reference toFIG. 12, an exemplary system for implementing the disclosed technology includes a general purpose computing device in the form of an exemplary conventional PC1200, including one or more processing units1202, a system memory1204, and a system bus1206that couples various system components including the system memory1204to the one or more processing units1202. The system bus1206may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The exemplary system memory1204includes read only memory (ROM)1208and random access memory (RAM)1210. A basic input/output system (BIOS)1212, containing the basic routines that help with the transfer of information between elements within the PC1200, is stored in ROM1208. As shown inFIG. 2, RAM1210can store computer-executable instructions for defining and coupling quantum circuits such as quantum circuits, with reduced basis transforms, shortened or combined Jordan-Wigner strings, or with nested quantum gates. In addition, some functions and procedures can be selected for implementation in conventional (non-quantum) computing hardware.

The exemplary PC1200further includes one or more storage devices1230such as a hard disk drive for reading from and writing to a hard disk, a magnetic disk drive for reading from or writing to a removable magnetic disk, and an optical disk drive for reading from or writing to a removable optical disk (such as a CD-ROM or other optical media). Such storage devices can be connected to the system bus1206by a hard disk drive interface, a magnetic disk drive interface, and an optical drive interface, respectively. The drives and their associated computer readable media provide nonvolatile storage of computer-readable instructions, data structures, program modules, and other data for the PC1200. Other types of computer-readable media which can store data that is accessible by a PC, such as magnetic cassettes, flash memory cards, digital video disks, CDs, DVDs, RAMs, ROMs, and the like, may also be used in the exemplary operating environment.

A number of program modules may be stored in the storage devices1230including an operating system, one or more application programs, other program modules, and program data. Storage of quantum syntheses and instructions for obtaining such syntheses can be stored in the storage devices1230. For example, nesting arrangements, reduced basis circuits, circuit series order, and CNOT gate cancellations can be defined by a quantum computer design application and circuit definitions can be stored for use in design. A user may enter commands and information into the PC1200through one or more input devices1240such as a keyboard and a pointing device such as a mouse. Other input devices may include a digital camera, microphone, joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the one or more processing units1202through a serial port interface that is coupled to the system bus1206, but may be connected by other interfaces such as a parallel port, game port, or universal serial bus (USB). A monitor1246or other type of display device is also connected to the system bus1206via an interface, such as a video adapter. Other peripheral output devices, such as speakers and printers (not shown), may be included. In some cases, a user interface is display so that a user can input a circuit for synthesis, and verify successful synthesis.

The PC1200may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer1260. In some examples, one or more network or communication connections1250are included. The remote computer1260may be another PC, a server, a router, a network PC, or a peer device or other common network node, and typically includes many or all of the elements described above relative to the PC1200, although only a memory storage device1262has been illustrated inFIG. 12. The personal computer1200and/or the remote computer1260can be connected to a logical a local area network (LAN) and a wide area network (WAN). Such networking environments are commonplace in offices, enterprise wide computer networks, intranets, and the Internet.

When used in a LAN networking environment, the PC1200is connected to the LAN through a network interface. When used in a WAN networking environment, the PC1200typically includes a modem or other means for establishing communications over the WAN, such as the Internet. In a networked environment, program modules depicted relative to the personal computer1200, or portions thereof, may be stored in the remote memory storage device or other locations on the LAN or WAN. The network connections shown are exemplary, and other means of establishing a communications link between the computers may be used.

With reference toFIG. 13, an exemplary system for implementing the disclosed technology includes computing environment1300, where compilation into quantum circuits is separated from quantum processing that uses the compiled circuits. The environment includes a quantum processing unit1302and one or more monitoring/measuring device(s)1346. The quantum processor executes quantum circuits that are precompiled by classical compiler unit1320using one or more classical processor(s)1310. The precompiled quantum circuits are downloaded into the quantum processing unit via quantum bus1306. In some cases, quantum circuits or portions thereof are predefined and stored as quantum circuit definitions in a memory1321. For example, nested or reduced basis quantum circuits associated with second quantization calculations can be stored in a library. A classical computer can be arranged to control a quantum computer or one or more quantum circuits thereof. The classical computer can receive the output of a classical or quantum computer. Based on the received output, the classical computer indicates which quantum circuits are to be used in subsequent quantum computations, provides definitions of suitable quantum circuits, or, in some cases, controls additional classical computations.

With reference toFIG. 13, the compilation is the process of translation of a high-level description of a quantum algorithm into a sequence of quantum circuits. Such high-level description may be stored, as the case may be, on one or more external computer(s)1360outside the computing environment1300utilizing one or more memory and/or storage device(s)1362, then downloaded as necessary into the computing environment1300via one or more communication connection(s)1350. The high-level description can be stored and interpreted classically, and a classical computer can control the sequence of gates defined in a quantum computer. The high level description also controls application of gates based on initial, intermediate, or final data values.

FIG. 14illustrates a representative design method1400. At1402, second quantization Hamiltonian terms are obtained, and at1404corresponding reduced Hamiltonian circuits are defined. As used herein, reduced Hamiltonian circuits refers to quantum circuits that include a control-Z gate associated with a Hamiltonian coefficient but lack at least one input side or output side basis change gate for at least one qubit corresponding to the associated spin orbitals. At1406, one or more CNOT gates in a Jordan Wigner string are cancelled, and additional basis change gates are canceled at1408. At1410selected gates are moved and in some cases, still more gates can be cancelled. At1412, a final circuit design is available.